Minimal Energy Local Systems on Curves

Charlie Wu

Abstract.

Let $\Sigma_{g,d}$ be an orientable topological surface of genus $g$ with $d$ punctures. When $g=0$ , Deroin and Tholozan studied the class of supra-maximal representations $\pi_{1}(\Sigma_{0,d})\to\operatorname{PSL}_{2}(\mathbb{R})$ , and they showed that the supra-maximal representations form a compact component of a real relative character variety. We study a collection of rank $n$ local systems on $\Sigma_{g,d}$ which we call of minimal energy. These are generalizations of supra-maximal representations, and underlie polarizable complex variations of Hodge structure for any choice of complex structure on $\Sigma_{g,d}$ . Like the supra-maximal representations, the minimal energy local systems form a compact component of a real relative character variety.

We show that when the local monodromy data around the punctures is chosen to be unitary and generic, and the relative character variety is nonempty, these minimal energy local systems always exist. When $g>0$ , we show that the minimal energy local systems come from unitary representations of $\pi_{1}(\Sigma_{g,d})$ . If $g=0$ we show that they do not always come from unitary representations, and we study their structure in general.

1. Introduction

Let $\Sigma_{g,d}$ be an orientable topological surface of genus $g$ and $d$ punctures. Let $C=(C_{1},\dots,C_{d})$ be a $d$ -tuple of conjugacy classes of $\operatorname{GL}_{n}(\mathbb{C})$ . We let $\chi(\pi_{1}(\Sigma_{g,d}),C)$ be the relative character variety of $\Sigma_{g,d}$ . This is the space of semi-simple rank $n$ representations of the fundamental group of $\Sigma_{g,d}$ such that the homotopy class of a simple closed loop $\gamma_{i}$ around some puncture $x_{i}$ lies in $C_{i}$ up to conjugation of the representation by $\operatorname{GL}_{n}(\mathbb{C})$ .

More precisely, we let the (relative) representation variety $\mathrm{Rep}(\pi_{1}(\Sigma_{g,d}),C)$ be given by

\displaystyle\mathrm{Rep}(\pi_{1}(X\setminus D),C)\subseteq\prod_{i=1}^{2g}\operatorname{GL}_{n}(\mathbb{C})\times\prod_{i=1}^{d}C_{i}

where $\mathrm{Rep}(\pi_{1}(\Sigma_{g,d}),C)$ is cut out by the relations

\displaystyle\prod_{i=1}^{2g}A_{i}=\prod_{i=1}^{d}B_{i}

for $A_{i}\in\operatorname{GL}_{n}(\mathbb{C})$ and $B_{i}\in C_{i}$ . Since the $C_{i}$ are algebraic subvarieties of $\operatorname{GL}_{n}(\mathbb{C})$ , $\mathrm{Rep}(\pi_{1}(\Sigma_{g,d}),C)$ inherits the structure of an algebraic variety. Then, $\operatorname{GL}_{n}(\mathbb{C})$ acts on $\mathrm{Rep}(\pi_{1}(\Sigma_{g,d}),C)$ by simultaneous conjugation of the $A_{i}$ ’s and $B_{i}$ ’s, and we define $\chi(\pi_{1}(\Sigma_{g,d}),C)$ to be $\mathrm{Rep}(\pi_{1}(\Sigma_{g,d}),C)//\operatorname{GL}_{n}(\mathbb{C})$ where the quotient is a GIT quotient.

1.1. Main results

We study a collection of local systems in $\chi(\pi_{1}(\Sigma_{g,n}),C)$ which we call of minimal energy. These local systems are defined in Definition 4.2, and they come from polarizable complex variations of Hodge structure ( $\mathbb{C}$ -VHS) satisfying a Hodge-theoretic vanishing condition. This definition a priori depends on a choice of complex structure for $\Sigma_{g,n}$ , but we show in Proposition 4.17 that this choice turns out to be auxiliary and does not matter.

Therefore, we give $\Sigma_{g,n}$ a complex structure and work with smooth algebraic curves. Let $X$ be a smooth proper curve of genus $g$ over the complex numbers and let $D=x_{1}+\dots+x_{d}$ be a reduced effective divisor on $X$ so that topologically, $\Sigma_{g,n}$ is homeomorphic to $X\setminus D$ . Suppose that the conjugacy classes $C_{1},\dots,C_{d}\subseteq\operatorname{GL}_{n}(\mathbb{C})$ each contain a unitary matrix, and as before we let $C=(C_{1},\dots,C_{d})$ .

The following results will be proven in Section 4, Section 5, and Section 6.

When $\chi(\pi_{1}(X\setminus D),C)$ is nonempty and smooth, we show that there is always a minimal energy local system in $\chi(\pi_{1}(X\setminus D),C)$ . Smoothness of $\chi(\pi_{1}(X\setminus D),C)$ is guaranteed provided that $C$ satisfies some genericity conditions (Definition 3.6).

Theorem 1.2.

If $\chi(\pi_{1}(X\setminus D),C)$ is nonempty and smooth, then there is a minimal energy local system in $\chi(\pi_{1}(X\setminus D),C)$ .

The methods used to prove Theorem 1.2 allow us conclude in Proposition 4.17 that the minimal energy local systems form almost all of the $\mathbb{C}$ -VHS lying in $\chi(\pi_{1}(X\setminus D),C)$ .

If $X$ has positive genus and our local monodromy data is generic, a minimal energy local system must come from a $\mathbb{C}$ -VHS with a one-step Hodge filtration. That is, these minimal energy local systems come from unitary representations. This is proven in Proposition 4.18. If the genus of $X$ is zero, then the minimal energy local systems do not have to come from a $\mathbb{C}$ -VHS with a one-step Hodge filtration. In Example 6.2, for any rank $r$ we find an example of a minimal energy local system on $\mathbb{P}^{1}\setminus D$ whose associated $\mathbb{C}$ -VHS has a two-step Hodge filtration. However, we show that when the number of punctures $d$ is very large compared to $r$ , this is the maximum number of pieces that can appear in the Hodge filtration of a $\mathbb{C}$ -VHS associated to a minimal energy local system.

Theorem 1.3.

Let $\mathbb{V}$ be an irreducible minimal energy local system of rank $n$ on $\mathbb{P}^{1}\setminus D$ with generic unitary local monodromy. If

\displaystyle\deg D\geq 4n^{2}-28

then $\mathbb{V}$ comes from a $\mathbb{C}$ -VHS with at most two steps in its Hodge filtration.

1.4. History and motivation

In [BG99], Robert Benedetto and William Goldman studied $\operatorname{SL}_{2}(\mathbb{C})$ -representations on $\Sigma_{0,4}$ . They observed that there is a compact component of a real relative character variety $\chi(\pi_{1}(X\setminus D),C)$ . Benedetto and Goldman first found this component by plotting the character variety on a computer.

Bertrand Deroin and Nicolas Tholozan, in [DT19], then studied $\operatorname{PSL}_{2}(\mathbb{R})$ -representations of $\pi_{1}(\Sigma_{0,d})$ when the number of punctures $d$ is at least three. They generalized the work of Benedetto-Goldman by showing that a class of representations, which they call supra-maximal, forms a compact component of a real relative character variety.

Tholozan and Jérémy Toulisse [TT21] generalized the work of Deroin-Tholozan to higher rank representations of $\pi_{1}(\Sigma_{0,3})$ . They show that relative $\operatorname{SU}(p,q)$ -character varieties admit compact components under certain conditions on the local monodromy when $d\geq 3$ , and they construct examples of nonempty $\operatorname{SU}(p,q)$ -character varieties satisfying these constraints.

The minimal energy local systems are generalizations of the supra-maximal representations (see [LLL23, Remark 4.3.4]), and they exist in any rank. One key feature of these minimal energy local systems is that, like the supra-maximal representations, they also form a compact component of a real relative character variety (see Proposition 4.19). This extends the work of Deroin-Tholozan and Tholozan-Toulisse, as the minimal energy local systems always exist (provided that the relative character variety is non-empty and smooth). Furthermore, the minimal energy local systems (and hence the compact components) exist for relative character varieties for all surfaces - not just punctured spheres.

1.5. Structure of the paper

Our methods heavily rely on Non-Abelian Hodge Theory, which gives us a correspondence between local systems on $X\setminus D$ and parabolic Higgs bundles on $X$ (with parabolic divisor $D$ ). In Section 2 and Section 3 we review the notions of parabolic bundles and $\mathbb{C}$ -VHS. In particular, we discuss how to take a $\mathbb{C}$ -VHS on $X\setminus D$ and produce a parabolic Higgs bundle.

Experts may wish to skip to Section 4, where we introduce the notion of minimal energy and prove Theorem 1.2 and Proposition 4.18. In Section 5 we prove Theorem 1.3, and in Section 6 we give Example 6.2 and give some applications of minimal energy local systems to Gromov-Witten theory.

1.6. Acknowledgements

I thank my advisor Daniel Litt. He suggested that I study these minimal energy local systems, and this project would not be possible without his encouragement and the many extremely helpful conversations I have had with him. I would also like to thank Simon Xu for the discussions I have had with him and for entertaining my many questions about Hodge theory. I am grateful to Samuel Bronstein and Arnaud Maret for their insightful questions and comments. I also thank Brian Collier as well as Bronstein and Maret for bringing the work of Tholozan and Toulisse to my attention.

2. Parabolic Bundles

We review the notions of parabolic bundles and parabolic sheaves. We use the notation in [LL24, Section 2]. Parabolic bundles are, roughly speaking, bundles carrying extra structure that keeps track of local unitary monodromy data.

Definition 2.1.

Let $E$ be a vector bundle on a smooth curve $C$ , and let $D=x_{1}+\dots+x_{d}$ be a reduced effective divisor on $C$ . A parabolic bundle $(E,\{E_{j}^{i}\},\{\alpha_{j}^{i}\})$ on $(C,D)$ is the data

(1)

a strictly decreasing filtration $E_{x_{j}}=E_{j}^{1}\supset E_{j}^{2}\supset\dots\supset E_{j}^{n_{j}+1}=0$ ,
(2)

a sequence of real numbers $0\leq\alpha_{j}^{1}<\alpha_{j}^{2}<\dots<\alpha_{j}^{n_{j}}<1$

for all $x_{j}$ appearing in $D$ . We write $E_{*}$ to mean $(E,\{E_{j}^{i}\},\{\alpha_{j}^{i}\})$ and we call $D$ the parabolic divisor of $E_{*}$ . For a fixed $x_{j}$ , we call $\{E_{j}^{i}\}$ the flag over $x_{j}$ and $\{\alpha_{j}^{i}\}$ the weights over $x_{j}$ . We call $\alpha_{j}^{i}$ the weight associated to $E_{j}^{i}$ .

Given a parabolic bundle $E_{*}$ , we get induced parabolic structures on subquotients of $E_{*}$ .

Definition 2.2.

Let $E_{*}$ be a parabolic bundle on $C$ with parabolic divisor $D$ , and let $F\subseteq E$ be a subbundle of $E$ . The vector bundles $F$ and $E/F$ carry an induced parabolic structure with parabolic divisor $D$ follows:

(1)

the flag of $F_{x_{j}}$ over $x_{j}$ is given by

$\displaystyle F_{x_{j}}=E_{j}^{1}\cap F_{x_{j}}\supseteq E_{j}^{2}\cap F_{x_{j}}\supseteq\dots\supseteq E_{j}^{n_{j}+1}\cap F_{x_{j}}=0$

after removing redundancies, and the weight associated to $F_{j}^{i}$ is given by $\max_{1\leq k\leq n_{j}}\{\alpha_{j}^{k}:F_{j}^{i}=E_{j}^{k}\cap F_{x_{j}}\}$ .

(2)

the flag of $(E/F)_{x_{j}}$ is given by

\displaystyle(E/F)_{x_{j}}=(E^{1}_{j}+F_{x_{j}})/F_{x_{j}}\supseteq(E^{2}_{j}+F_{x_{j}})/F_{x_{j}}\supseteq\dots\supseteq(E^{n_{j}}_{j}+F_{x_{j}})/F_{x_{j}}=0

after removing redundancies, and the weight associated to $(E/F)_{x_{j}}^{i}$ is given by $\max_{1\leq k\leq n_{j}}\{\alpha_{j}^{k}:(E/F)_{x_{j}}^{i}=(E_{j}^{k}+F_{x_{j}})/F_{x_{j}}\}$ .

Definition 2.3.

Let $E_{*}$ be a parabolic vector bundle on $C$ with parabolic divisor $D=x_{1}+\dots+x_{d}$ . The parabolic degree is the quantity

\displaystyle\operatorname{par-deg}E_{*}

\displaystyle:=\deg(E)+\sum_{j=1}^{d}\sum_{i=1}^{n_{j}}\alpha_{j}^{i}\cdot\dim(E_{j}^{i}/E_{j}^{i+1})

and we call $\mu_{*}(E_{*}):=\operatorname{par-deg}E_{*}/\rank E$ to be the parabolic slope of $E_{*}$ .

Parabolic degree satisfies properties similar to those satisfied by the usual notion of degree, and there is a corresponding notion of parabolic stability.

Lemma 2.4.

Let $0\to F_{*}\to E_{*}\to Q_{*}\to 0$ be a short exact sequence of parabolic bundles. Then $\operatorname{par-deg}(E_{*})=\operatorname{par-deg}(F_{*})+\operatorname{par-deg}(Q_{*})$ . A parabolic bundle $E_{*}$ is semi-stable (resp. stable) if and only if for every quotient bundle $Q_{*}$ of $E_{*}$ , $\mu_{*}(E_{*})\leq\mu_{*}(Q_{*})$ (resp. $\mu_{*}(E_{*})<\mu_{*}(Q_{*})$ ).

Proof.

See [LL24, Lemma 2.4.5]. ∎

To define the notion of parabolic tensor product and homomorphisms between parabolic bundles, we will need the notion of a parabolic sheaf. To that end, let $\mathbb{R}$ be the category consisting of real numbers with a single morphism $i^{a,b}:a\to b$ if $a\leq b$ .

Definition 2.5.

An $\mathbb{R}$ -filtered $\mathscr{O}_{X}$ -module is a functor $E:\mathbb{R}\to\mathrm{Mod}_{\mathscr{O}_{X}}$ where $\mathrm{Mod}_{\mathscr{O}_{X}}$ is the category of $\mathscr{O}_{X}$ -modules. We write $E_{\alpha}$ for $E(\alpha)$ and we write $E_{*}$ for the functor $E$ . Write $i_{E}^{a,b}=E(i^{a,b})$ .

Set $E[\alpha]_{*}$ to be the functor given by $E[\alpha]_{\beta}=E_{\alpha+\beta}$ , and $i^{a,b}_{E[\alpha]}=i^{a+\alpha,b+\alpha}_{E}$ . We write $i^{[\alpha,\beta]}_{E}:E[\alpha]_{*}\to E[\beta]_{*}$ to be the natural transformation so that $i_{E}^{[\alpha,\beta]}(\gamma)=i_{E}^{\alpha+\gamma,\beta+\gamma}$ . If $f:E_{*}\to F_{*}$ is a natural transformation of $\mathbb{R}$ -filtered $\mathscr{O}_{X}$ -modules, we write $f[\alpha]:E[\alpha]_{*}\to F[\alpha]_{*}$ to be the induced map.

Definition 2.6.

A parabolic sheaf $E_{*}$ with divisor $D$ is an $\mathbb{R}$ -filtered $\mathscr{O}_{X}$ -module with an isomorphism $j_{E}:E_{*}(-D)\to E_{*}[1]$ such that $i_{E}^{[0,1]}\circ j_{E}=\operatorname{id}_{E_{*}}\otimes i_{D}$ where $i_{D}:\mathscr{O}_{X}(-D)\to\mathscr{O}_{X}$ is the inclusion.

A natural transformation $f:E_{*}\to F_{*}$ is a parabolic morphism if $f[1]\circ j_{E}=(f\otimes\operatorname{id})\circ j_{F}$ . Let $\operatorname{Hom}(E_{*},F_{*})$ be the set of parabolic morphisms and $\mathscr{H}\kern-2.0ptom(E_{*},F_{*})$ to be the sheaf of parabolic morphisms defined by $\mathscr{H}\kern-2.0ptom(E_{*},F_{*})(U)=\operatorname{Hom}({E_{*}}|_{U},{F_{*}}|_{U})$ . The sheaf $\mathscr{H}\kern-2.0ptom(E_{*},F_{*})$ has a parabolic structure given by $\mathscr{H}\kern-2.0ptom(E_{*},F_{*})_{\alpha}=\mathscr{H}\kern-2.0ptom(E_{*},F[\alpha]_{*})$ with morphism $\mathscr{H}\kern-2.0ptom(E_{*},F_{*})_{\alpha}\to\mathscr{H}\kern-2.0ptom(E_{*},F_{*})_{\beta}$ induced by $F[\alpha]_{*}\to F[\beta]_{*}$ .

The notion of parabolic vector bundle and parabolic sheaf are related.

Example 2.7.

Let $(E,\{E_{j}^{i}\},\{\alpha_{j}^{i}\}$ be a parabolic vector bundle with parabolic divisor $D$ . For $0\leq\alpha<1$ , write $\beta(\alpha,j)=\min\{j:\alpha_{j}^{i}\geq\alpha\}$ for all $\alpha\leq\max\{\alpha_{j}^{i}\}$ and $\beta(\alpha,j)=n_{j}+1$ for $\max\{\alpha_{j}^{i}\}<\alpha<1$ . Then set

\displaystyle E_{\alpha}

\displaystyle=\bigcap_{j=1}^{n}\ker(E\to E_{x_{j}}/E_{j}^{\beta(\alpha,j)}).

We extend outside $[0,1)$ by $E_{x+n}=E_{x}(-nD)$ for all $n\in\mathbb{Z}$ , with morphisms $i^{\alpha,\beta}:E_{\alpha}\to E_{\beta}$ to be their natural inclusions.

Definition 2.8.

Let $E_{*}$ and $F_{*}$ be two parabolic sheaves with the same parabolic divisor $D$ , and let $j:C\setminus D\hookrightarrow C$ the natural inclusion. Suppose that $E_{\alpha}$ and $F_{\alpha}$ are both locally free for all $\alpha$ . Then, we define the parabolic tensor product $E_{*}\otimes F_{*}$ to be the parabolic sheaf such that $(E_{*}\otimes F_{*})_{\alpha}$ is generated by all $E_{a}\otimes E_{b}$ where $a+b=\alpha$ when viewed as a subbundle of $j_{*}j^{*}(E\otimes F)$ , and morphisms $i_{E_{*}\otimes F_{*}}^{\alpha,\beta}$ to be the natural inclusion map.

A parabolic sheaf $E_{*}$ is constant along intervals $[\alpha_{i},\alpha_{i+1})$ . To each parabolic sheaf, we can associate a parabolic sheaf $\widehat{E}_{*}$ which is constant along $(\alpha_{i},\alpha_{i+1}]$ . (See [BY96, Figure 1]).

Definition 2.9.

Let $E_{*}$ be a parabolic sheaf. We define $\widehat{E}_{*}$ by the rule

\displaystyle\widehat{E}_{x}

\displaystyle=\begin{cases}E_{x}&x\neq\alpha_{i}\\ E_{\alpha_{i+1}}&x=\alpha_{i},\end{cases}

and we call $\widehat{E}_{*}$ the coparabolic sheaf associated to $E_{*}$ . If $E_{*}$ is a parabolic bundle, then we call $\widehat{E}_{*}$ the coparabolic bundle associated to $E_{*}$ .

Proposition 2.10.

If we write $F_{*}^{\vee}$ to mean $\mathscr{H}\kern-2.0ptom(F_{*},\mathscr{O}_{X})_{*}$ (where $\mathscr{O}_{X}$ has the trivial parabolic structure), then $\mathscr{H}\kern-2.0ptom(E_{*},F_{*})_{*}\cong(E_{*}^{\vee}\otimes F_{*})_{*}$ .

Proof.

See [Yok95, Lemma 3.6]. ∎

If $E_{*}$ and $F_{*}$ are rank $1$ parabolic bundles, then $\mathscr{H}\kern-2.0ptom(E_{*},F_{*})\cong E^{\vee}\otimes F(-D^{\prime})$ where $D^{\prime}\subseteq D$ is the divisor consisting of points $x_{j}$ where the weight of $E_{*}$ is greater than the weight of $F_{*}$ . Similarly, $\widehat{\mathscr{H}\kern-2.0ptom(E_{*},F_{*})}\cong E^{\vee}\otimes F(-D^{\prime\prime})$ where $D^{\prime\prime}$ is the divisor of points where the weight of $E_{*}$ is greater than or equal to the weight of $F_{*}$ .

Definition 2.11.

A parabolic Higgs bundle $(E_{*},\theta)$ on $(C,D)$ is a parabolic bundle $E_{*}$ on $(C,D)$ along with an $\operatorname{End}(E_{*})$ -valued logarithmic $1$ -form $\theta\in H^{0}(X,\widehat{\operatorname{End}(E_{*})\otimes\Omega_{X}^{1}(\log D)})$ . The map $\theta$ is called a Higgs field.

There are notions of stability and semi-stability for parabolic Higgs bundles.

Definition 2.12.

If $(E_{*},\theta)$ is a Higgs bundle with a parabolic structure, then we say that $(E_{*},\theta)$ is a parabolic Higgs bundle is stable (resp. semi-stable) if all sub-Higgs bundles $(F_{*},\theta|_{F})$ satisfy $\mu_{*}(F_{*})<\mu_{*}(E_{*})$ (resp. $\mu_{*}(F_{*})\leq\mu_{*}(E_{*})$ ).

Lemma 2.13.

If $F_{*}$ and $G_{*}$ two parabolic bundles on $\mathbb{P}^{1}$ , then

(2.1)

\displaystyle\operatorname{par-deg}\mathscr{H}\kern-2.0ptom(E_{*},F_{*})_{*}=\rank E\cdot\operatorname{par-deg}F_{*}-\rank F\cdot\operatorname{par-deg}E_{*}.

Proof.

See [Bis03, Section 2] for a discussion on computing $\operatorname{par-deg}(E_{*}\otimes F_{*})_{*}$ for $E_{*}$ and $F_{*}$ parabolic bundles. Then to get the statement for $\mathscr{H}\kern-2.0ptom(E_{*},F_{*})_{*}$ we note that $\mathscr{H}\kern-2.0ptom(E_{*},F_{*})_{*}=((E_{*})^{\vee}\otimes F_{*})_{*}$ . ∎

Lemma 2.14.

Let $E_{*}$ be a parabolic vector bundle with parabolic divisor $D$ . Then

\displaystyle\deg E\leq\operatorname{par-deg}E_{*}<\deg E+\rank E\cdot\deg D.

Proof.

By Definition 2.3

\displaystyle\operatorname{par-deg}E_{*}=\deg E+\sum_{x_{j}\in D}\sum_{i=1}^{n_{j}}\alpha_{i}^{j}\cdot\dim(E_{j}^{i}/E_{j}^{i+1}).

Since our parabolic weights $\alpha_{i}^{j}$ lie in $[0,1)$ ,

\displaystyle 0\leq\sum_{x_{j}\in D}\sum_{i=1}^{n_{j}}\alpha_{i}^{j}\cdot\dim(E_{j}^{i}/E_{j}^{i+1})

\displaystyle<\sum_{x_{j}\in D}\sum_{i=1}^{n_{j}}\dim(E_{j}^{i}/E_{j}^{i+1})=\sum_{x_{j}\in D}\dim(E_{x_{j}}).

But $E$ is a vector bundle so the fibers have dimension equal to $\rank E$ . Therefore

\displaystyle\deg V\leq\operatorname{par-deg}V_{*}<\deg V+\sum_{x_{j}\in D}\dim(E_{x_{j}})=\deg V+\rank E\cdot\deg D.

∎

We seek to study parabolic Higgs bundles on $(\overline{X},D)$ because they correspond to local systems on $\overline{X}\setminus D$ . This correspondence is work of Simpson in [Sim90], and we include it as Proposition 2.15 for convenience. We will refer to Proposition 2.15 below as the Non-Abelian Hodge Theorem.

Proposition 2.15.

Let $X=\overline{X}\setminus D$ where $\overline{X}$ is a smooth proper curve and $D$ is a reduced effective divisor. There is a one-to-one correspondence between parabolic semi-stable Higgs bundles of parabolic degree $0$ on $\overline{X}$ with parabolic divisor $D$ and the local systems on $X$ .

Proof.

See [Sim90, Theorem on page 718]. ∎

We will describe which Higgs bundles correspond to local systems underlying a complex variation of Hodge structure in Section 3.

3. Variations of Hodge Structure

We give background on variations of Hodge structure and their associated Higgs bundles in this section. We primarily use the notation of [LL24, Section 4].

Definition 3.1.

Let $X$ be a smooth irreducible variety over $\mathbb{C}$ . A complex variation of Hodge struture ( $\mathbb{C}$ -VHS) is a triple $(V,V^{p,q},D)$ where $V$ is a complex $C^{\infty}$ -bundle on $X$ , $\oplus V^{p,q}=V$ is a direct sum decomposition, and $D$ is a flat connection satisfying

\displaystyle D(V^{p,q})\subseteq A^{1,0}(V^{p,q})\oplus A^{0,1}(V^{p,q})\oplus A^{1,0}(V^{p-1,q+1})\oplus A^{0,1}(V^{p+1,q-1}).

A polarization on $(V,V^{p,q},D)$ is a flat Hermitian form $\psi$ such that the $V^{p,q}$ are orthogonal to eachother under $\psi$ , and that on each $V^{p,q}$ the form $(-1)^{p}\psi$ is positive-definite. A polarizable $\mathbb{C}$ -VHS is a $\mathbb{C}$ -VHS which admits a polarization $\psi$ .

Given a $\mathbb{C}$ -VHS $(V,V^{p,q},D)$ we call the holomorphic flat bundle $(E,\nabla):=(\ker(D)\otimes_{\mathbb{C}}\mathscr{O},\operatorname{id}\otimes d)$ the holomorphic flat bundle associated to the $\mathbb{C}$ -VHS. We get a filtration $F^{p}E$ on $E$ induced by the filtration $F^{p}V=\oplus_{j\geq p}V^{j,q}$ on $V$ , known as the Hodge filtration. The filtration satisfies Griffiths transversality:

\displaystyle\nabla(F^{p})\subseteq F^{p-1}\otimes\Omega_{X}^{1}(\log D).

If $\mathbb{V}$ is a a local system isomorphic to $\ker(D)$ for some polarizable $\mathbb{C}$ -VHS $(V,V^{p,q},D)$ , then we say that $\mathbb{V}$ underlies a polarizable $\mathbb{C}$ -VHS.

Definition 3.2.

Let $X=\overline{X}\setminus D$ with $D$ a simple normal crossings divisor, and $(E,\nabla)$ a flat holomorphic vector bundle on $X$ . The Deligne canonical extension of $(E,\nabla)$ is the unique logarithmic flat bundle $(\overline{E},\overline{\nabla})$ on $\overline{X}$ with regular singularities along $D$ , along with an isomorphism $(\overline{E},\overline{\nabla})|_{X}\cong(E,\nabla)$ , and such that the eigenvalues of the residues of $\overline{\nabla}$ along $D$ have real part in $[0,1)$ .

The existence of the Deligne canonical extension is a theorem of Deligne [Del70, Remarques 5.5(i)].

Definition 3.3.

Let $X=\overline{X}\setminus D$ be a curve where $\overline{X}$ is some smooth proper curve and $D$ is a reduced effective divisor. Let $(E,\nabla)$ be a flat holomorphic vector bundle of rank $n$ on $X$ with regular singularities along $D$ . Suppose at some $x_{j}$ , the eigenvalues of the residue matrix $\Res(\nabla)(x_{j})$ are given by $\eta_{j}^{1},\dots,\eta_{j}^{n}$ . Then, the real parts $\real(\eta_{j}^{i})$ lie in $[0,1)$ . We write $\alpha_{j}^{i}=\real(\eta_{j}^{i})$ and order the $\alpha_{j}^{i}$ in increasing order and remove repetitions so

\displaystyle 0\leq\alpha_{j}^{1}<\dots<\alpha_{j}^{n_{j}}<1.

Let $E_{j}^{i}\subseteq E_{x_{j}}$ be the sum of all of the generalized eigenspaces of $\Res(\nabla)(x_{j})$ such that the real part of the associated eigenvalues are at least $\alpha_{j}^{i}$ . Then, $E=(E,\{E_{j}^{i}\},\{\alpha_{j}^{i}\})$ is the parabolic bundle associated to the connection $\nabla$ .

Definition 3.4.

Let $\overline{X}$ be a smooth projective curve, $Z$ a reduced effective divisor, and $X=\overline{X}\setminus Z$ . Let $(V,V^{p,q},D)$ be a $\mathbb{C}$ -VHS on $X$ and $(E,\nabla)$ its associated holomorphic flat bundle with Hodge filtration $F^{\bullet}$ . Let $(\overline{E}_{*},\overline{\nabla})$ be its associated parabolic bundle on $X$ in the sense of Definition 3.2 and Definition 3.3. Then by [Bru17, Section 7], there is a canonical extension of the Hodge filtration $F^{\bullet}$ to $(\overline{E},\overline{\nabla})$ . Let $\overline{F}^{\bullet}$ be this extension.

We call $(\oplus_{p}\operatorname{Gr}_{\overline{F}^{\bullet}}^{p}\overline{E}_{*},\oplus\operatorname{Gr}_{\overline{F}^{\bullet}}^{p}\overline{\nabla})$ the parabolic Higgs bundle associated to the variation of Hodge structure. The $\mathscr{O}_{X}$ -linearity of the map $\oplus\operatorname{Gr}_{\overline{F}^{\bullet}}^{p}\overline{\nabla}$ is due to Griffiths transversality of $\overline{\nabla}$ and $\overline{F}^{\bullet}$ .

Proposition 3.5.

Let $X=\overline{X}\setminus Z$ where $\overline{X}$ is a smooth proper curve and $Z$ is a reduced effective divisor. Let $(V,V^{p,q},D)$ be a complex variation of Hodge structure on $X$ , $(E,F^{\bullet},\nabla)$ its associated holomorphic flat bundle with Hodge filtration $F^{\bullet}$ , and $(\oplus E^{p}_{*},\oplus\theta_{p}):=(\oplus_{p}\operatorname{Gr}_{\overline{F}^{\bullet}}^{p}\overline{E}_{*},\oplus\operatorname{Gr}_{\overline{F}^{\bullet}}^{p}\overline{\nabla})$ be its associated parabolic Higgs bundle defined in Definition 3.4. Then,

(1)

$\oplus E_{*}^{p}$ is of parabolic degree zero,
(2)

$(\oplus E_{*}^{p},\oplus\theta_{p})$ is parabolic semi-stable as a Higgs bundle,
(3)

if the local system $\ker(\nabla)$ is irreducible, then $(\oplus E_{*}^{p},\oplus\theta_{p})$ is parabolic stable as a Higgs bundle.

Proof.

This is due to Simpson [Sim90, Theorem 5]. ∎

Proposition 3.5 produces a parabolic Higgs bundle from a $\mathbb{C}$ -VHS. The parabolic Higgs bundle is the same as the one given by Simpson’s Non-Abelian Hodge Theorem (Proposition 2.15).

Definition 3.6.

Let $X$ be a smooth projective curve and $D=x_{1}+\dots+x_{d}$ be a reduced effective divisor on $X$ . We fix a positive integer $n$ . Let $\{\alpha_{j}^{i}\}$ be a collection of parabolic weights.

(1)

We say that $\{\alpha_{j}^{i}\}$ is smooth with respect to $(X,D,n)$ if every semi-stable parabolic Higgs bundle of rank $n$ on $(X,D)$ of parabolic degree $0$ with the weights $\{\alpha_{j}^{i}\}$ is actually stable.
(2)

We say that the collection of weights $\{\alpha_{j}^{i}\}$ is distinct with respect to $(X,D,n)$ if for all $j$ , $n_{j}=n$ where $n_{j}$ is the number of weights over $x_{j}$ .
(3)

We say that the weights $\{\alpha_{j}^{i}\}$ are generic with respect to $(X,D,n)$ if (1) and (2) are both satisfied.

Let $\mathbb{V}$ be a local system on $X\setminus D$ with unitary local monodromy around the punctures. We say the local monodromy of $\mathbb{V}$ is generic if the collection of associated weights of the associated parabolic Higgs bundle is generic (via the Non-Abelian Hodge Theorem).

Example 3.7.

Let $(X,D,n)$ be fixed. Let $\{\alpha_{j}^{i}\}\subseteq[0,1)$ be a set of size $\deg D\cdot n$ so that no proper subset of the $\{\alpha_{j}^{i}\}$ has elements which sum to an integer. Then, $\{\alpha_{j}^{i}\}$ is generic with respect to $(X,D,n)$ in the sense of Definition 3.4 (3). To check this, we take any parabolic semi-stable Higgs bundle $(E_{*},\theta)$ of parabolic degree $0$ and observe that any sub-Higgs bundle $(F_{*},\theta|_{F})$ of $(E_{*},\theta)$ satisfies

\displaystyle\frac{\operatorname{par-deg}F_{*}}{\rank F}=\mu_{*}(F_{*})\leq\mu_{*}(E_{*})=\frac{\operatorname{par-deg}E_{*}}{\rank E}=0.

Then since

(3.1)

\displaystyle\operatorname{par-deg}F_{*}=\deg F+\sum_{j=1}^{\deg D}\sum_{k=1}^{\rank F}\alpha_{j}^{{i_{k}}}

where $\{\alpha_{j}^{i_{k}}\}$ is the collection of weights of $F_{*}$ . We note that

\displaystyle\deg F+\sum_{j=1}^{\deg D}\sum_{k=1}^{\rank F}\alpha_{j}^{{i_{k}}}<0

because by assumption, the expression

\displaystyle\sum_{j=1}^{\deg D}\sum_{k=1}^{\rank F}\alpha_{j}^{{i_{k}}}

is not an integer and $\deg F$ is always an integer.

Let $X$ be a curve. Given a parabolic Higgs bundle $(E_{*},\theta)$ on $(X,D)$ , the $i$ -th Higgs cohomology group of $(E,\theta)$ is defined to be

\displaystyle\mathbb{H}^{i}\left(E\xrightarrow{\theta}\widehat{E\otimes\Omega_{X}^{1}(\log D)}\right).

If $(E_{*},\theta)$ underlies a $\mathbb{C}$ -VHS of weight $w$ , $E=\oplus_{p+q=w}E^{p,q}$ then the Higgs cohomology has a Hodge structure of weight $w+i$ with $(p,q)$ part given by

\displaystyle\mathbb{H}^{p,q}\left(E\xrightarrow{\theta}\widehat{E\otimes\Omega_{X}^{1}(\log D)}\right)

\displaystyle:=\mathbb{H}^{i}\left(E^{p,q-i}\xrightarrow{\theta}\widehat{E^{p-1,q+i-1}\otimes\Omega_{X}^{1}(\log D)}\right).

This means that

\displaystyle\mathbb{H}^{i}\left(E\xrightarrow{\theta}\widehat{E\otimes\Omega_{X}^{1}(\log D)}\right)=\bigoplus_{p+q=w+i}\mathbb{H}^{i}\left(E^{p,q-i}\xrightarrow{\theta}\widehat{E^{p-1,q+i-1}\otimes\Omega_{X}^{1}(\log D)}\right).

Definition 3.8.

Let $(E_{*},\theta)$ be a parabolic Higgs bundle underlying a $\mathbb{C}$ -VHS on a curve $X$ with parabolic divisor $D$ . We say $\mathbb{H}^{1}(E_{*}\to\widehat{E_{*}\otimes\Omega_{X}^{1}(\log D)}\to\dots)$ is of Hodge length $\ell$ if $\mathbb{H}^{k,1-k}(E_{*}\to\widehat{E_{*}\otimes\Omega_{X}^{1}(\log D)})=0$ for all $k<-\ell+1$ and $k>\ell$ .

4. Minimal Energy

Let $X$ be a smooth proper curve. As before, we let $M=M(X,D,n,\{\alpha_{i}^{j}\})$ be the coarse moduli space of semi-stable parabolic Higgs bundles of rank $n$ of parabolic degree $0$ on $X$ with parabolic divisor $D$ and parabolic weights $\{\alpha_{i}^{j}\}$ . This space is constructed in [Yok93, Section 2]. As $(X,D,n)$ are fixed, any smoothness, distinctness, and genericity conditions on a collection of weights $\{\alpha_{j}^{i}\}$ are assumed to be with respect to $(X,D,n)$ in the sense of Definition 3.6.

As before, for a semi-stable parabolic Higgs field $(E_{*},\theta)$ to be in $M$ , we require that $\theta\in H^{0}(X,\widehat{\operatorname{End}(E)_{*}\otimes\Omega_{X}^{1}(\log D)})$ . This ensures that when we deform a Higgs bundle corresponding to a local system with unitary local monodromy, the resulting Higgs bundle gives us a local system with unitary local monodromy.

Remark 4.1.

If $(E_{*},\theta)=(\oplus E^{p},\oplus\theta_{p})$ is a stable parabolic Higgs bundle underlying an (irreducible) $\mathbb{C}$ -VHS, then the adjoint Higgs bundle $\operatorname{Ad}(E_{*})$ also underlies a $\mathbb{C}$ -VHS. This has a canonical grading, where the $k$ -th graded piece $\operatorname{Ad}(E_{*})^{k,-k}$ is given by $\oplus\mathscr{H}\kern-2.0ptom(E^{p}_{*},E^{p+k}_{*})_{*}$ . Then,

\displaystyle\mathbb{H}^{1}=\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\xrightarrow{\theta}\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})

has a Hodge structure of weight $1$ . In this case $\dim_{\mathbb{C}}\mathbb{H}^{p,1-p}=\dim_{\mathbb{C}}\mathbb{H}^{1-p,p}$ because $\operatorname{Ad}(E_{*})$ is self dual so $\mathbb{H}^{1}$ has a real Hodge structure. This is because $\operatorname{Ad}(E_{*})$ is polarized, so it is dual to its conjugate. But it is self-dual, so its conjugate is itself.

We now define the notion of minimal energy.

Definition 4.2.

Let $(E_{*},\theta)$ be a stable parabolic Higgs bundle on a smooth proper curve $X$ with parabolic divisor $D$ and parabolic weights $\{\alpha_{i}^{j}\}$ corresponding to a complex variation of Hodge structure. We say that $(E_{*},\theta)$ is an irreducible minimal energy parabolic Higgs bundle (or is of minimal energy) if the Higgs cohomology group

\displaystyle\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})

of $M$ at $(E_{*},\theta)$ is of Hodge length $1$ .

If $(V,V^{p,q},D^{\prime})$ is the $\mathbb{C}$ -VHS corresponding to the minimal energy Higgs bundle $(E_{*},\theta)$ , then we say that $(V,V^{p,q},D^{\prime})$ is a minimal energy variation of Hodge structure.

Similarly, if $\mathbb{V}$ is a local system on $X\setminus D$ with unitary local monodromy such that the corresponding parabolic Higgs bundle $(E_{*},\theta)$ (under the Non-Abelian Hodge correspondence) is of minimal energy, then we say $\mathbb{V}$ is a minimal energy local system (or is of minimal energy).

The notion of minimal energy is actually a condition on the tangent space of $M$ at $(E_{*},\theta)$ .

Remark 4.3.

If $(E_{*},\theta)$ is a smooth point of $M$ , then the tangent space at $(E_{*},\theta)$ is isomorphic to $\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})$ . If $(E_{*},\theta)$ is stable, then it is a smooth point in $M$ .

Proof.

See [Tha02, Section 3] for the smoothness claim. A detailed proof that the tangent space at $(E_{*},\theta)$ is isomorphic $\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})$ to can be found in [Bot95, Proposition 5.2.1].

∎

By Remark 4.3, if our parabolic weights are chosen smoothly (in the sense of Definition 3.6), then every semi-stable parabolic Higgs bundle with those weights is actually stable. In this case, our moduli space $M$ is smooth.

Remark 4.4.

In the non-parabolic setting, there is an energy functional on the moduli space of Higgs bundles given by $\norm{(E,\theta)}=i\int_{X}\tr(\theta\wedge\theta^{*})$ where $\theta^{*}$ is the adjoint to $\theta$ with respect to a harmonic metric. If $(E,\theta)$ is stable, then this metric is unique. Collier and Wentworth [CW19, Section 4.1] showed that stable Higgs bundles which minimize $\norm{(E,\theta)}$ are those whose tangent space is of Hodge length $1$ by exploiting the fact that the energy functional is a Morse-Bott function on the moduli space of Higgs bundles [Hit87, Section 8]. The tangent space to the moduli space of Higgs bundles splits into a direct sum $\oplus_{j\in\mathbb{Z}}T_{j}$ where $T_{j}=\mathbb{H}^{j,1-j}$ [Hit92, Section 8]. On each piece $T_{j}$ , the Hessian of $\norm{\cdot}$ has eigenvalue $j$ and hence a Higgs bundle cannot minimize the energy functional unless $T_{j}=0$ for $j<0$ . Symmetry of the decomposition of the tangent space implies that the only nonzero pieces are $T_{0}$ and $T_{1}$ which is precisely the Hodge length $1$ condition described in Definition 4.2.

There are some classes of local systems which are always of minimal energy.

Example 4.5.

A unitary local system $\mathbb{V}$ corresponds to a parabolic Higgs bundle underlying a $\mathbb{C}$ -VHS coming in one piece $E_{*}=E^{1}_{*}$ and Higgs field $\theta=0$ . This is a consequence of the Mehta-Seshadri correspondence for irreducible unitary local systems and stable parabolic bundles [MS80, Theorem 4.1]. Then, $\mathbb{V}$ is a minimal energy local system.

Example 4.6.

A rigid local system is always of minimal energy.

We give an example of a rigid local system. A result of Katz [Kat96, Theorem 1.1.2] says that for a rank $n$ irreducible local system $\mathbb{V}$ on $\mathbb{P}^{1}\setminus\{p_{1},\dots,p_{k}\}$ with local monodromy data (conjugate to) $(A_{1},\dots,A_{k})$ , $\mathbb{V}$ is rigid if and only if

\displaystyle(2-k)n^{2}+\sum_{i=1}^{k}\dim Z(A_{i})=2

where $Z(A_{i})$ is the centralizer of $A_{i}$ .

In the case where $k=3$ and $n=2$ , Katz’s condition requires that

\displaystyle\dim Z(A_{1})+\dim Z(A_{2})+\dim Z(A_{3})=6.

For a $2\times 2$ non-scalar matrix $B$ , $\dim Z(B)=2$ . Hence, any irreducible rank $2$ local system on $\mathbb{P}^{1}\setminus\{0,1,\infty\}$ with non-scalar local monodromy around the punctures must be rigid and therefore is of minimal energy.

There is a a $\mathbb{G}_{m}$ -action on $M$ . If $t\in\mathbb{G}_{m}$ and $(E_{*},\theta)\in M$ then we act on $(E_{*},\theta)$ by $t$ by $t\cdot(E_{*},\theta)=(E_{*},t\theta)$ . Simpson [Sim90, Theorem 8] shows that the fixed points of this action on $M$ is precisely the locus of parabolic Higgs bundles coming from a $\mathbb{C}$ -VHS under the correspondence in Proposition 2.15.

The $\mathbb{G}_{m}$ -action on $M$ induces a $\mathbb{G}_{m}$ -action the tangent space $\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})$ , and this $\mathbb{G}_{m}$ -action interacts with the Hodge structure in a natural way.

Proposition 4.7.

The $\mathbb{G}_{m}$ -action on $M$ induces a $\mathbb{G}_{m}$ -action on the tangent space of $M$ at a fixed point $(E_{*},\theta)$ . This induces a $\mathbb{G}_{m}$ -action (through the isomorphism in Remark 4.3) on $\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})$ . The action of $t\in\mathbb{G}_{m}$ on each piece of the Hodge structure $\mathbb{H}^{p,1-p}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})$ is given by $t^{p}$ .

Proof.

Let $(E_{*},\theta)$ be a fixed point. By [Sim90, Theorem 8], $(E_{*},\theta)$ comes from a $\mathbb{C}$ -VHS and hence has a graded Higgs bundle $(E_{*},\theta)=(\oplus E^{p}_{*},\oplus\theta^{p})$ . The grading is unique up to shifting. Once we pick a grading, we get a map $f:(\oplus E^{p}_{*},\oplus t\theta^{p})\to(\oplus E^{p}_{*},\oplus\theta^{p})$ given by $f|_{E^{p}}=t^{p}$ . This is an isomorphism of parabolic Higgs bundles.

The map $f$ naturally gives us a morphism of complexes

where $\psi$ is the induced Higgs field on $\operatorname{Ad}(E_{*})$ . Here, $\tilde{f}|_{\operatorname{Ad}(E_{*})^{p,-p}}$ is given by multiplication by $t^{p}$ . Hence, we get a map on hypercohomology

\displaystyle\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D))\to\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D))

induced by $\tilde{f}$ .

In order to compute the $\mathbb{G}_{m}$ -action on $\mathbb{H}^{p,1-p}$ we consider the morphism of complexes

Let $I^{\bullet}$ be a resolution of $\operatorname{Ad}(E_{*})^{p,-p}$ and $J^{\bullet}$ be a resolution of $\widehat{\operatorname{Ad}(E_{*})^{p-1,1-p}\otimes\Omega_{X}^{1}(\log D)}$ . Then,

\displaystyle\mathbb{H}^{p,1-p}=H^{1}(\Gamma(I^{0})\to\Gamma(I^{1})\oplus\Gamma(J^{0})\to\Gamma(I^{2})\oplus\Gamma(J^{1})\to\dots)

and we can compute the $\mathbb{G}_{m}$ -action on $\mathbb{H}^{p,1-p}$ using the above description. Here, we are implicitly using the isomorphism in Remark 4.3.

The complex

is resolved by ${I^{\bullet}}\to{J^{\bullet}}^{\prime}$ where ${J^{\bullet}}^{\prime}$ and $J^{\bullet}$ are related by multiplication by $t$ . This is because we have an isomorphism of complexes

To obtain the chain map, we can compose the vertical maps to get

The chain map is given by multiplication by $t^{p}$ , so $\mathbb{G}_{m}$ acts on $\mathbb{H}^{p,1-p}$ by multiplication by $t^{p}$ . ∎

Theorem 1.2 says that the energy Higgs bundles always exist when the moduli space $M$ is smooth, and a generic parabolic Higgs bundle in $M$ can be deformed to one of minimal energy. We prove this by using the parabolic Hitchin map defined in [Yok93, Page 495] as well as Bialynicki-Birula theory. We briefly review the definitions and properties here.

4.8. The Hitchin map

Definition 4.9.

The parabolic Hitchin map is the map $h:M\to\oplus_{j=1}^{n}H^{0}(X,\Omega_{X}^{1}(\log D)^{\otimes j})$ defined by

\displaystyle h((E_{*},\theta))=(\tr(\wedge^{1}\theta),\tr(\wedge^{2}\theta),\dots,\tr(\wedge^{n}\theta)).

The map $h$ sends a parabolic Higgs bundle to the coefficients of the characteristic polynomial of its Higgs field.

Proposition 4.10.

The parabolic Hitchin map is proper.

Proof.

See [Yok93, Theorem 5.10] where they prove that the valuative criterion for properness holds. ∎

Lemma 4.11.

Let $(E_{*},\theta)$ be a stable parabolic Higgs bundle on $X$ . Then, $\lim_{t\to 0}(E_{*},t\theta)$ exists and underlies a $\mathbb{C}$ -VHS. In particular, $(E_{*},\theta)$ can be deformed to a parabolic Higgs bundle underlying a $\mathbb{C}$ -VHS.

Proof.

This is proven in [Moc06, Proposition 1.9(3)]. ∎

4.12. The Bialynicki-Birula stratification

Let $G=\mathbb{G}_{m}$ , $Z$ be some smooth variety over $\mathbb{C}$ with an action of $G$ on $Z$ , and $Z^{G}$ the locus of fixed points of the $G$ -action. Suppose $Z$ can be covered by $G$ -invariant quasi-affine opens. Then, Bialynicki-Birula [BB73, Theorem 4.1 and Theorem 4.3] proves the following:

Theorem 4.13.

Let $Z^{G}=\cup_{i=1}^{r}(Z^{G})_{i}$ be the decomposition of $Z^{G}$ into connected components. Then for any $i=1,\dots,r$ , there exists are unique locally closed non-singular $G$ -invariant subschemes $Z_{i}^{+}$ and $Z_{i}^{-}$ and unique morphisms $\gamma_{i}^{+}:Z_{i}^{+}\to(Z^{G})_{i}$ and $\gamma_{i}^{-}:Z_{i}^{-}\to(Z^{G})_{i}$

(a)

$(Z^{G})_{i}$ is a closed subscheme of $Z_{i}^{+}$ (resp. $Z_{i}^{-}$ ) and $\gamma_{i}^{+}|_{(Z^{G})_{i}}$ (resp. $\gamma_{i}^{-}|_{(Z^{G})_{i}}$ ) is the identity.
(b)

$Z_{i}^{+}$ (resp. $Z_{i}^{-}$ ) with its induced action of $G$ and with the map $\gamma_{i}^{+}$ (resp. $\gamma_{i}^{-}$ ) is a $G$ -fibration over $(Z^{G})_{i}$ .
(c)

For any closed $a\in(Z^{G})_{i}$ , we have the following equality of tangent spaces $T_{a}(Z_{i}^{+})=T_{a}(Z)^{0}\oplus T_{a}(Z)^{+}$ and $T_{a}(Z_{i}^{-})=T_{a}(Z)^{0}\oplus T_{a}(Z)^{-}$ where for a $G$ -module $V$ , $V^{0}$ is the $G$ -invariant submodule of $V$ , $V^{+}$ is spanned by the elements $v\in V$ such that $t\cdot v=t^{k}v$ for $k>0$ , and $V^{-}$ is spanned by the elements $v\in V$ such that $t\cdot v=t^{k}v$ for $k<0$ .

If furthermore $\lim_{t\to 0}z\cdot t$ exists for all $z\in Z$ , then

(d)

$(Z_{i}^{+})^{G}=(Z^{G})_{i}$ (resp. $(Z_{i}^{-})_{G}=(Z_{G})_{i}$ ) for $i=1,\dots,r$ .
(e)

$Z_{i}^{+}$ (resp. $Z_{i}^{-}$ ) with the map $\gamma_{i}^{+}$ (resp. $\gamma_{i}^{-}$ ) is a $G$ -fibration over $(Z^{G})_{i}$
(f)

For any closed $a\in(Z^{G})_{i}$ , $T_{a}(Z_{i}^{+})=T_{a}((Z^{G})_{i})\oplus T_{a}(Z_{i}^{+})$ (resp. $T_{a}(Z_{i}^{-})=T_{a}((Z^{G})_{i})\oplus T_{a}(Z_{i}^{-})$ )

Remark 4.14.

The statements of Theorem 4.13 (d), (e), and (f) in [BB73, Theorem 4.3] require that our variety $Z$ is complete. This is to ensure that $\lim_{t\to 0}z\cdot t$ exists, so requiring the weaker assumption that $\lim_{t\to 0}z\cdot t$ exists is sufficient.

Theorem 4.13 gives us a stratification of $M$ (when it is smooth) compatible with the $\mathbb{G}_{m}$ -action on $M$ . This is because smoothness of $M$ over $\mathbb{C}$ ensures that $M$ is a normal variety with a $\mathbb{G}_{m}$ -action, so by [Sum74, Corollary 2] we can cover $M$ with $\mathbb{G}_{m}$ -invariant affine open subsets of $M$ . By Lemma 4.11, $\lim_{t\to 0}(E_{*},t\theta)$ exists for all $(E_{*},\theta)$ so parts (d), (e), and (f) of Theorem 4.13 also apply. We can realize $M$ as a fibration over the locus of $\mathbb{G}_{m}$ -fixed points, which by [Sim90, Theorem 8] is precisely the $\mathbb{C}$ -VHS locus.

4.15. Proof of Theorem 1.2

We are now ready to prove that if $M$ is smooth, then minimal energy parabolic Higgs bundles exist. By Simpson’s Non-Abelian Hodge Theorem (Proposition 2.15), this is the same as showing the minimal energy local systems in $\chi(\pi_{1}(X\setminus D),C)$ exist.

Proof of Theorem 1.2.

We use the Bialynicki-Birula stratification (Theorem 4.13). By assumption $M$ is smooth, and Lemma 4.11 implies that $\lim_{t\to 0}(E_{*},t\theta)$ exists so all parts of Theorem 4.13 hold in our setting. The $\mathbb{G}_{m}$ -fixed locus inside $M$ is precisely the $\mathbb{C}$ -VHS locus by [Sim90, Theorem 8]. Let $Y$ be this locus, and $Y=Y_{1}\cup\dots\cup Y_{r}$ its decomposition into irreducible components. By Theorem 4.13, there are locally closed non-singular $\mathbb{G}_{m}$ -invariant subschemes $M_{i}^{+}$ and morphisms $\gamma_{i}^{+}:M_{i}^{+}\to Y_{i}$ that $Y_{i}$ is inside $M_{i}^{+}$ and $\gamma_{i}^{+}$ restricts to the identity on $Y_{i}$ . Assume without loss of generality that $M_{1}^{+}$ has the greatest dimension among the $M_{i}^{+}$ for $i=1,\dots,r$ . Note that $\dim M_{1}^{+}=\dim M$ .

Let $y\in Y_{1}$ . We claim that $y$ is of minimal energy. We define the following subspaces of $T_{M,y}$ :

	$\displaystyle W^{-}$	$\displaystyle=\mathrm{span}\{v\in T_{M,y}:t\cdot v=t^{k}v\text{ for some }k<0\}$
	$\displaystyle W^{0}$	$\displaystyle=\mathrm{span}\{v\in T_{M,y}:t\cdot v=v\}$
	$\displaystyle W^{+}$	$\displaystyle=\mathrm{span}\{v\in T_{M,y}:t\cdot v=t^{k}v\text{ for some }k>0\}$

By Proposition 4.7, $W^{-}\cong\oplus_{p<0}\mathbb{H}^{p,1-p}$ , $W^{0}\cong\mathbb{H}^{0,1}$ , and $W^{+}\cong\oplus_{p>0}\mathbb{H}^{p,1-p}$ . Since $\dim_{\mathbb{C}}\mathbb{H}^{p,1-p}=\dim_{\mathbb{C}}\mathbb{H}^{1-p,p}$ for all $p\in\mathbb{Z}$ (see Remark 4.1), to show that $\mathbb{H}^{p,1-p}=\mathbb{H}^{1-p,p}=0$ for $p>1$ it is enough it is enough to show that $W^{-}=0$ since

\displaystyle\dim W^{-}=\dim\left(\bigoplus_{p<0}\mathbb{H}^{p,1-p}\right)=\dim\left(\bigoplus_{p>1}\mathbb{H}^{p,1-p}\right).

By parts (c) and (f) of Theorem 4.13, we know that $T_{M_{1}^{+},y}=W^{0}\oplus W^{+}$ . But since $\dim M_{1}^{+}=\dim M$ , this implies that

\displaystyle W^{0}\oplus W^{+}=T_{M_{1}^{+},y}=T_{M,y}=W^{-}\oplus W^{0}\oplus W^{+}.

But then, $W^{-}=0$ and we are done. ∎

4.16. Some properties of minimal energy local systems

The minimal energy local systems make up “most” of the $\mathbb{C}$ -VHS in $M$ . We make this precise below.

Proposition 4.17.

Suppose $M$ is smooth, and let $Y$ be the locus of $\mathbb{C}$ -VHS inside $M$ . A $\mathbb{C}$ -VHS $p\in Y$ is of minimal energy if and only if it lies in the irreducible component of $Y$ of largest dimension. The (complex) dimension of this irreducible component is half of the (complex) dimension of $M$ .

Proof.

By Proposition 4.7 and Remark 4.3, the tangent space of $M$ at a $\mathbb{C}$ -VHS $p\in M$ is of the form $T_{M,p}=\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})=\oplus_{p}\mathbb{H}^{p,1-p}$ and the action of $\mathbb{G}_{m}$ on $\mathbb{H}^{p,1-p}$ is given by multiplication by $t^{p}$ . Hence, parts (c) and (f) of Theorem 4.13 imply that $\mathbb{H}^{0,1}$ is isomorphic $T_{Y,p}$ (which is the tangent space of $Y$ at $p$ ).

Since $\mathbb{H}^{p,1-p}\cong\mathbb{H}^{1-p,p}$ as vector spaces for all $p\in\mathbb{Z}$ , this implies that

\displaystyle\dim Y=\dim T_{Y,p}=\dim\mathbb{H}^{0,1}\leq\frac{1}{2}\dim T_{M,p}=\frac{1}{2}\dim M

with equality if and only if $p$ is of minimal energy. Since we know by Theorem 1.2 that minimal energy $\mathbb{C}$ -VHS always exist, we know that the top dimensional component of $Y$ is always half of the dimension of $M$ .

If $q=(F_{*},\omega)\in Y$ lies in component of smaller dimension $Y^{\prime}\subseteq Y$ , then

\displaystyle\dim Y^{\prime}=\dim T_{Y,q}=\dim\mathbb{H}^{0,1}(\operatorname{Ad}(F_{*})\to\widehat{\operatorname{Ad}(F_{*})\otimes\Omega_{X}^{1}(\log D)})<\frac{1}{2}\dim T_{M,q}=\frac{1}{2}\dim M

which implies that there is some $k>1$ such that $\mathbb{H}^{k,1-k}(\operatorname{Ad}(F_{*})\to\widehat{\operatorname{Ad}(F_{*})\otimes\Omega_{X}^{1}(\log D)})$ is nonzero which violates the definition of minimal energy. ∎

If $X$ is a positive genus curve, then the minimal energy parabolic Higgs bundles must correspond to unitary representations of $\pi_{1}(X\setminus D)$ . We make this precise below in Proposition 4.18.

Proposition 4.18.

Let $X$ be a smooth proper curve of genus $g>0$ , and $D$ a reduced effective divisor. If $(E_{*},\theta)$ is a stable parabolic Higgs bundle of minimal energy (with parabolic divisor $D$ ) on $X$ of parabolic degree $0$ with distinct parabolic weights $\{\alpha_{i}^{j}\}$ , then the Higgs field is zero (and so $(E_{*},\theta)$ comes from a unitary representation).

Proof.

We write $E_{*}=E^{r}_{*}\oplus\dots\oplus E^{1}_{*}$ where the graded pieces of our Higgs fields $\theta^{p}:E^{p}\to E^{p-1}\otimes\Omega_{X}^{1}(\log D)$ are all nonzero. Since $(E_{*},\theta)$ is stable, it is a smooth point of the moduli space of semi-stable parabolic Higgs bundles of parabolic degree $0$ on $X$ with parabolic divisor $D$ and parabolic weights $\{\alpha_{i}^{j}\}$ . Therefore by Remark 4.3, the tangent space to $M$ at $(E_{*},\theta)$ is isomorphic to $\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})$ .

If $\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})$ is of Hodge length $1$ , then we require that

\displaystyle\mathbb{H}^{1}(0\to\mathscr{H}\kern-2.0ptom(E^{1}_{*},E^{r}_{*})_{*}\otimes\Omega_{X}^{1}(\log D))=\mathbb{H}^{1}(\mathscr{H}\kern-2.0ptom(E^{r},E^{1})_{*}\to 0)=0.

Then, Riemann-Roch for curves yields

	$\displaystyle 0$	$\displaystyle\leq h^{0}(X,\mathscr{H}\kern-2.0ptom(E^{r}_{},E^{1}_{})_{})-h^{1}(X,\mathscr{H}\kern-2.0ptom(E^{r}_{},E^{1}_{})_{})$
		$\displaystyle=\deg\mathscr{H}\kern-2.0ptom(E^{r}_{},E^{1}_{})_{}+\rank\mathscr{H}\kern-2.0ptom(E^{r}_{},E^{1}_{})_{}(1-g).$

Therefore

\displaystyle 1-g

\displaystyle\geq-\frac{\deg\mathscr{H}\kern-2.0ptom(E^{r}_{*},E^{1}_{*})_{*}}{\rank\mathscr{H}\kern-2.0ptom(E^{r}_{*},E^{1}_{*})_{*}}=-\mu_{*}(\mathscr{H}\kern-2.0ptom(E^{r}_{*},E^{1}_{*})_{*}).

Since $\mathscr{H}\kern-2.0ptom(E^{r}_{*},E^{1}_{*})_{*}$ has parabolic degree at most $0$ and

\displaystyle\deg\mathscr{H}\kern-2.0ptom(E^{r}_{*},E^{1}_{*})_{*}\leq\operatorname{par-deg}\mathscr{H}\kern-2.0ptom(E^{r}_{*},E^{1}_{*})_{*}\leq 0

by Lemma 2.14, this implies that $1-g=0$ if and only if $\deg\mathscr{H}\kern-2.0ptom(E^{r}_{*},E^{1}_{*})_{*}=\operatorname{par-deg}\mathscr{H}\kern-2.0ptom(E^{r}_{*},E^{1}_{*})_{*}=0$ . Otherwise, $-\mu_{*}(\mathscr{H}\kern-2.0ptom(E^{r}_{*},E^{1}_{*})_{*})>0$ and so $g=0$ .

If $\deg\mathscr{H}\kern-2.0ptom(E^{r}_{*},E^{1}_{*})_{*}=\operatorname{par-deg}\mathscr{H}\kern-2.0ptom(E^{r}_{*},E^{1}_{*})_{*}=0$ then that implies that $E^{r}=E^{1}$ since we have assumed that the parabolic weights on $(E_{*},\theta)$ are all distinct at each point in our parabolic divisor. But then, we know that $E_{*}=E^{r}_{*}=E^{1}_{*}$ and so the Higgs field on $E_{*}$ is zero. The Non-Abelian Hodge Theorem (Proposition 2.15) tells us that $(E_{*},\theta)$ must come from a unitary representation. ∎

The author thanks Daniel Litt for sharing the following argument learned from a conversation with Bertrand Deroin and Nicolas Tholozan.

Proposition 4.19.

Let $X$ be a smooth proper curve and $D$ a reduced effective divisor on $X$ . Let $(E_{*},\theta)=(\oplus E^{k}_{*},\oplus\theta_{k})$ be a minimal parabolic energy Higgs bundle of rank $n$ on $X$ with parabolic divisor $D$ and generic parabolic weights. If $p=\sum\rank_{k\in\mathbb{Z}}E^{2k}$ and $q=\sum_{k\in\mathbb{Z}}\rank E^{2k+1}$ , then $p+q=n$ and the locus of minimal energy Higgs bundles form a compact component of the $\operatorname{SU}(p,q)$ -relative character variety.

Proof.

Let $\mathbb{V}$ be a local system corresponding to a polarizable $\mathbb{C}$ -VHS. Then, the representation $\rho$ corresponding to $\mathbb{V}$ preserves the polarization, and the polarization is a type $(p,q)$ -Hermitian form. Hence, $\rho$ is an $\operatorname{SU}(p,q)$ -representation of $\pi_{1}(X\setminus D)$ . The $\mathbb{C}$ -VHS correspond to parabolic Higgs bundles, and the parabolic Higgs bundles they correspond to have nilpotent Higgs field. Hence, the parabolic Higgs bundles coming from $\mathbb{C}$ -VHS lie in $h^{-1}(0)$ where $h$ is the parabolic Hitchin map (Definition 4.9).

We now consider the minimal energy local systems. As minimal energy local systems underlie a $\mathbb{C}$ -VHS, they necessarily correspond to $\operatorname{SU}(p,q)$ -representations and also lie in $h^{-1}(0)$ . Proposition 4.10 tells us that $h$ is proper, so $h^{-1}(0)$ is compact. Let $Y$ be the $\operatorname{SU}(p,q)$ -relative character variety. Then, $Y$ is a real form of the $\operatorname{GL}_{n}(\mathbb{C})$ -relative character variety because $\operatorname{SU}(p,q)$ is a real form of $\operatorname{GL}_{n}(\mathbb{C})$ . Hence, $Y$ is closed inside the $\operatorname{GL}_{n}(\mathbb{C})$ -relative character variety. Let $Y^{\prime}\subseteq M$ be the locus of parabolic Higgs bundles in $M$ corresponding to $Y$ . We know that $Y^{\prime}$ is closed in $M$ , so $Y^{\prime}\cap h^{-1}(0)$ is compact as it is a closed subset of $h^{-1}(0)$ which is compact.

Let $Z\subseteq Y^{\prime}\cap h^{-1}(0)$ be the locus of minimal energy parabolic Higgs bundles. It suffices to show that $Z$ is a component of $Y$ , as then $Z$ would be closed in $Y$ and hence a compact component of $Y$ . We know by Proposition 4.17 that $\frac{1}{2}\dim M=\dim Z\leq\dim Y^{\prime}\cap h^{-1}(0)\leq\dim Y=\frac{1}{2}\dim M$ . Therefore, $\dim Z=\dim Y$ which implies that $Z$ is a component of $Y$ . ∎

This mirrors the lower-rank case in [DT19, Proposition 2.6] where the Deroin-Tholozan representations (which correspond to the minimal energy local systems in rank $2$ ), form a compact component of the real character variety.

5. Genus $g=0$

We restrict our attention to minimal energy parabolic Higgs bundles on $\mathbb{P}^{1}$ . In contrast to the higher genus case (see Proposition 4.18) where the minimal energy local systems correspond to unitary representations, the minimal energy Higgs bundles do not necessarily underlie a $\mathbb{C}$ -VHS with only one graded piece.

Let $(E_{*},\theta)$ be a parabolic Higgs bundle on $\mathbb{P}^{1}$ with distinct parabolic weights and parabolic divisor $D$ corresponding to an irreducible representation. Since $(E_{*},\theta)$ underlies a $\mathbb{C}$ -VHS it is of the form $(\oplus_{p=1}^{r}E^{p}_{*},\oplus\theta^{p})$ where our Higgs field $\theta^{p}:E^{p}\to E^{p-1}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)$ is nonzero. Suppose there $r$ pieces, where $r>2$ . We define $V^{k}=\oplus_{p}\mathscr{H}\kern-2.0ptom(E^{p}_{*},E^{p+k}_{*})_{*}=\operatorname{Ad}(E_{*})^{k,-k}$ , and $\theta_{k}:V^{k}\to V^{k-1}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)$ to be the induced Higgs field on the $k$ -th graded piece of $\operatorname{Ad}(E_{*})$ .

Our ultimate goal in this section is to prove Theorem 1.3. We work with minimal energy parabolic Higgs bundles $(E_{*},\theta)$ , and we bound the number of graded pieces in $(E_{*},\theta)$ . This bounds the number of steps in the Hodge filtration of the $\mathbb{C}$ -VHS associated to $(E_{*},\theta)$ .

We first prove the following intermediate bound.

Proposition 5.1.

Let $(E_{*},\theta)$ be a minimal energy parabolic Higgs bundle on $\mathbb{P}^{1}$ of with at least $3$ graded pieces and with generic unitary local monodromy data. If $(E_{*},\theta)$ underlies an irreducible representation, then

\displaystyle\deg D\leq\frac{2\rank V^{r-1}-\rank V^{1}+2\sum_{k=2}^{r-1}\rank V^{k-1}+\sum_{k=2}^{r-1}(2k-3)\rank V^{k}}{\sum_{k=2}^{r-1}(\rank V^{k-1}-\rank\mathrm{coker}\,\theta_{k})+\sum_{k=2}^{r-1}(k-2)\rank V^{k}}.

This bound will be used to prove Theorem 1.3. The bound in Theorem 1.3, which is independent of the ranks of the individual graded pieces, is obtained by bounding the $\rank V^{k}$ in terms of $\rank E_{*}$ and applying Proposition 5.1.

To that end, we prove a series of lemmas which place restrictions the behavior of the Higgs field on $(\operatorname{Ad}(E_{*}),\oplus\theta_{k})$ whenever $(E_{*},\theta)$ is of minimal energy.

Lemma 5.2.

If $(E_{*},\theta)$ is a parabolic Higgs bundle of minimal energy with distinct parabolic weights, then for all $1<k<r$

\displaystyle H^{1}(\mathbb{P}^{1},\ker\theta_{k})=H^{0}(\mathbb{P}^{1},\mathrm{coker}\,\theta_{k})=0

Proof.

By Definition 4.2 and Remark 4.3, $\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\operatorname{Ad}(E_{*})\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D))$ is of Hodge length $1$ . Note that since our parabolic weights are distinct, $\widehat{V^{k}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)}=V^{k}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)$ for all $1\leq k\leq r$ . Therefore, the complex $\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)}$ has associated graded pieces

	$\displaystyle 0$	$\displaystyle\to V^{r-1}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)$
	$\displaystyle V^{r-1}$	$\displaystyle\to V^{r-2}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)$
		$\displaystyle\vdots$
	$\displaystyle V^{2-r}$	$\displaystyle\to V^{1-r}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)$
	$\displaystyle V^{1-r}$	$\displaystyle\to 0.$

The graded pieces of $\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\operatorname{Ad}(E_{*})\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D))$ are given by

\displaystyle\mathbb{H}^{k,1-k}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})=\mathbb{H}^{1}(V^{k}\to V^{k-1}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D))

whenever $k\neq 1$ . We have an exact sequence of complexes of sheaves

\displaystyle 0\to\ker\theta_{k}[0]\to[V^{k}\to V^{k-1}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)]\to\mathrm{coker}\,\theta_{k}[-1]\to 0

so we get a long exact sequence on hypercohomology

Since $\mathbb{H}^{0}(\mathbb{P}^{1},\mathrm{coker}\,\theta_{k}[-1])=0$ , $\mathbb{H}^{1}(\mathbb{P}^{1},\mathrm{coker}\,\theta_{k}[-1])=H^{0}(\mathbb{P}^{1},\mathrm{coker}\,\theta_{k})$ , and $H^{1}(\mathbb{P}^{1},\ker\theta_{k}[0])=H^{1}(\mathbb{P}^{1},\ker\theta_{k})$ we have a short exact sequence

\displaystyle 0\to H^{1}(\mathbb{P}^{1},\ker\theta_{k})\to\mathbb{H}^{1}(V^{k}\to V^{k-1}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D))\to H^{0}(\mathbb{P}^{1},\mathrm{coker}\,\theta_{k})\to 0

and so $\mathbb{H}^{k,1-k}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})=\mathbb{H}^{1}(V^{k}\to V^{k-1}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D))$ vanishes if and only if $H^{1}(\mathbb{P}^{1},\ker\theta_{k})=H^{0}(\mathbb{P}^{1},\mathrm{coker}\,\theta_{k})=0$ . Since $(E_{*},\theta)$ is of minimal energy, $\mathbb{H}^{k,1-k}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})=0$ for $1<k<r$ . ∎

This lemma gives us strong conditions on the Higgs fields of minimal energy Higgs bundles.

Corollary 5.3.

Let $(E_{*},\theta)$ be a minimal energy parabolic Higgs bundle on $\mathbb{P}^{1}$ . Then for $1<k<r$ , $\mathrm{coker}\,\theta_{k}$ is a locally free sheaf.

Proof.

As $\mathbb{P}^{1}$ is a smooth curve and $\mathrm{coker}\,\theta_{k}$ is a coherent sheaf, we obtain a splitting $\mathrm{coker}\,\theta_{k}=(\mathrm{coker}\,\theta_{k})_{\mathrm{lf}}\oplus(\mathrm{coker}\,\theta_{k})_{\mathrm{tors}}$ where $(\mathrm{coker}\,\theta_{k})_{\mathrm{lf}}$ is locally free and $(\mathrm{coker}\,\theta_{k})_{\mathrm{tors}}$ is a torsion sheaf. Since $(E_{*},\theta)$ is a minimal energy parabolic Higgs bundle, $\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})$ is of Hodge length $1$ . Lemma 5.2 implies that for $1<k<r$ ,

\displaystyle 0=H^{0}(\mathbb{P}^{1},\mathrm{coker}\,\theta_{k})=H^{0}(\mathbb{P}^{1},(\mathrm{coker}\,\theta_{k})_{\mathrm{lf}})\oplus H^{0}(\mathbb{P}^{1},(\mathrm{coker}\,\theta_{k})_{\mathrm{tors}}).

But $(\mathrm{coker}\,\theta_{k})_{\mathrm{tors}}$ is a torsion sheaf on a curve and is hence supported at finitely many points. Therefore $H^{0}(\mathbb{P}^{1},(\mathrm{coker}\,\theta_{k})_{\mathrm{tors}})=0$ if and only if $(\mathrm{coker}\,\theta_{k})_{\mathrm{tors}}$ has empty support if and only if $(\mathrm{coker}\,\theta_{k})_{\mathrm{tors}}=0$ . Therefore $\mathrm{coker}\,\theta_{k}=(\mathrm{coker}\,\theta_{k})_{\mathrm{lf}}$ as desired. ∎

Corollary 5.4.

If $(E_{*},\theta)$ is a minimal energy parabolic Higgs bundle on $\mathbb{P}^{1}$ , then for $1<k<r$ , $\deg\mathrm{coker}\,\theta_{k}\leq-\rank\mathrm{coker}\,\theta_{k}$ and $\deg\ker\theta_{k}\geq-\rank\ker\theta_{k}$ .

Proof.

As $(E_{*},\theta)$ is a minimal energy Higgs bundle on $\mathbb{P}^{1}$ , we have $H^{0}(\mathbb{P}^{1},\mathrm{coker}\,\theta_{k})=H^{1}(\mathbb{P}^{1},\ker\theta_{k})=0$ by Lemma 5.2 whenever $1<k<r$ .

As $\ker\theta_{k}$ is a vector bundle on $\mathbb{P}^{1}$ , $\ker\theta_{k}\cong\mathscr{O}(a_{1})\oplus\dots\oplus\mathscr{O}(a_{\rank\ker\theta_{k}})$ where $a_{j}$ are integers and $a_{1}+\dots+a_{\rank\ker\theta_{k}}=\deg\ker\theta_{k}$ . Therefore

\displaystyle H^{1}(\mathbb{P}^{1},\ker\theta_{k})=H^{1}(\mathbb{P}^{1},\mathscr{O}(a_{1}))\oplus\dots\oplus H^{1}(\mathbb{P}^{1},\mathscr{O}(a_{\rank\ker\theta_{k}}))=0

if and only if all of the integers $a_{j}$ are at least $-1$ . So, $\deg\ker\theta_{k}=a_{1}+\dots+a_{\rank\ker\theta_{k}}\geq-\rank\ker\theta_{k}$ .

By Corollary 5.3, $\mathrm{coker}\,\theta_{k}$ is a vector bundle and is isomorphic to $\mathscr{O}(b_{1})\oplus\dots\oplus\mathscr{O}(b_{\rank\mathrm{coker}\,\theta_{k}})$ . Since

\displaystyle H^{0}(\mathbb{P}^{1},\mathrm{coker}\,\theta_{k})=H^{0}(\mathbb{P}^{1},\mathscr{O}(b_{1}))\oplus\dots\oplus H^{0}(\mathbb{P}^{1},\mathscr{O}(b_{\rank\mathrm{coker}\,\theta_{k}}))=0

we know that all of the integers $b_{j}$ are at most $-1$ . Therefore

\displaystyle\deg\mathrm{coker}\,\theta_{k}=b_{1}+\dots+b_{\rank\mathrm{coker}\,\theta_{k}}\leq-\rank\mathrm{coker}\,\theta_{k}.

∎

We now prove a series of technical lemmas used in the proof of Proposition 5.1.

Lemma 5.5.

If $(E,\theta)$ is of minimal energy on $\mathbb{P}^{1}$ , then

\displaystyle\operatorname{par-deg}V^{1}

\displaystyle\geq(\deg D-1)\sum_{k=2}^{r-1}(k-2)\rank V^{k}-\sum_{k=2}^{r-1}(k-1)\rank V^{k}

and $\deg V^{r-1}=\rank V^{r-1}\cdot(1-\deg D)$ .

Proof.

Since $(E,\theta)$ is of minimal energy, $\mathbb{H}^{1}(V^{1-r}\to 0)=H^{1}(\mathbb{P}^{1},V^{1-r})=0$ . But observe that $V^{1-r}$ is a sub-parabolic Higgs bundle of $\operatorname{Ad}(E_{*})$ with zero Higgs field, so $V^{1-r}$ must be a semi-stable vector bundle on $\mathbb{P}^{1}$ and by Lemma 2.13 and Lemma 2.14

\displaystyle\deg V^{1-r}\leq\operatorname{par-deg}V^{1-r}=\operatorname{par-deg}E^{1}\cdot\rank E^{r}-\operatorname{par-deg}E^{r}\cdot\rank E^{1}<0.

Therefore $V^{1-r}\cong\mathscr{O}(-1)^{\oplus\rank V^{1-r}}$ . Equivalently, $V^{r-1}\cong(\mathscr{O}(1)^{\oplus\rank V^{r-1}})(-D)$ and so $\deg V^{r-1}=\rank V^{r-1}\cdot(1-\deg D)$ .

We have short exact sequences

\displaystyle 0\to\ker\theta_{k}\to V^{k}\to V^{k-1}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)\to\mathrm{coker}\,\theta_{k}\to 0

and for $r<k<2$ we have by Corollary 5.4

	$\displaystyle\deg V^{k-1}$	$\displaystyle=\deg V^{k}+\rank V^{k-1}\cdot(2-\deg D)+\deg\mathrm{coker}\,\theta_{k}-\deg\ker\theta_{k}$
		$\displaystyle\leq\deg V^{k}+\rank V^{k-1}\cdot(2-\deg D)+\rank\ker\theta_{k}-\rank\mathrm{coker}\,\theta_{k}.$

Since $\rank\ker\theta_{k}-\rank\mathrm{coker}\,\theta_{k}=\rank V^{k}-\rank V^{k-1}$ , we then have that

\displaystyle\deg V^{k-1}

\displaystyle\leq\deg V^{k}+\rank V^{k-1}\cdot(1-\deg D)+\rank V^{k}

By induction, for $\ell>0$ we obtain the bound

\displaystyle\deg V^{k-\ell}

\displaystyle\leq\deg V^{k}+(1-\deg D)\cdot\sum_{j=1}^{\ell}\rank V^{k-j}+\sum_{j=1}^{\ell}\rank V^{k-j+1}.

In the special case where $k=r-1$ , we have by substituting for $\deg V^{r-1}$

\displaystyle\deg V^{r-1-\ell}

\displaystyle\leq(1-\deg D)\cdot\sum_{j=0}^{\ell}\rank V^{r-1-j}+\sum_{j=1}^{\ell}\rank V^{r-j}.

We now observe that $V^{0}\oplus V^{-1}\oplus\dots\oplus V^{1-r}$ is a sub-Higgs bundle, so $\operatorname{par-deg}(V^{0}\oplus V^{-1}\oplus\dots\oplus V^{1-r})\leq 0$ by semi-stability of $\operatorname{Ad}(E_{*})$ . Equivalently, $\operatorname{par-deg}(V^{r-1}\oplus\dots\oplus V^{1})\geq 0$ . We first bound $\operatorname{par-deg}V^{k}$ . By Lemma 2.14 and our above bounds on $\deg V^{k}$ , we have

\displaystyle\operatorname{par-deg}V^{k}

\displaystyle\leq\sum_{j=k}^{r-1}\rank V^{j}+(1-\deg D)\cdot\sum_{j=k+1}^{r-1}\rank V^{j}.

Since $\operatorname{par-deg}(V^{r-1}\oplus\dots\oplus V^{1})\geq 0$ we have that

	$\displaystyle\operatorname{par-deg}V^{1}$	$\displaystyle\geq-\sum_{k=2}^{r-1}\operatorname{par-deg}V^{k}$
		$\displaystyle\geq-\sum_{k=2}^{r-1}\left(\sum_{j=k}^{r-1}\rank V^{j}+(1-\deg D)\cdot\sum_{j=k+1}^{r-1}\rank V^{j}\right),$

so we have obtained a lower bound in terms of the $\rank V^{k}$ . Simplifying yields

\displaystyle\operatorname{par-deg}V^{1}

\displaystyle\geq(\deg D-1)\sum_{k=2}^{r-1}(k-2)\rank V^{k}-\sum_{k=2}^{r-1}(k-1)\rank V^{k}.

∎

Lemma 5.6.

For all $1<k<r$ ,

\displaystyle\rank V^{k-1}-\rank\mathrm{coker}\,\theta_{k}\geq r-k.

Proof.

Since $V^{k-1}\otimes\Omega_{X}^{1}(\log D)$ surjects onto $\mathrm{coker}\,\theta_{k}$ , we know that $\rank V^{k-1}-\rank\mathrm{coker}\,\theta_{k}\geq 0$ . By Lemma 5.4 we have

\displaystyle\rank V^{k-1}-\rank\mathrm{coker}\,\theta_{k}=\rank V^{k}-\rank\ker\theta_{k}

and hence if $\rank V^{k-1}-\rank\mathrm{coker}\,\theta_{k}=0$ , then $\theta_{k}$ must be the zero map.

We show that this is not the case. The map $\theta_{k}:\oplus_{j=1}^{r-k}\mathscr{H}\kern-2.0ptom(E^{j}_{*},E^{j+k}_{*})_{*}\to\oplus_{j=1}^{r-k+1}\mathscr{H}\kern-2.0ptom(E^{j}_{*},E^{j+k-1}_{*})_{*}$ is given by

	$\displaystyle\theta_{k}(f_{1}^{1+k},\dots,f_{r-k}^{r})$
	$\displaystyle=(\theta^{1+k}\circ f_{1}^{1+k},\theta^{k+2}\circ f_{2}^{k+2}-(f_{1}^{1+k}\otimes\operatorname{id})\circ\theta^{2}\dots,$
	$\displaystyle\qquad\theta^{r}\circ f_{r-k}^{r}-(f_{r-k-1}^{r-1})\otimes\operatorname{id})\circ\theta^{r-k},-(f_{r-k}^{k}\otimes\operatorname{id})\circ\theta^{r-k+1})$

where $f_{j}^{j+k}\in\mathscr{H}\kern-2.0ptom(E^{j}_{*},E^{j+k}_{*})_{*}$ .

As $\mathrm{coker}\,\theta_{k}$ is a locally free sheaf for all $1<k<r$ by Corollary 5.3, we can compute $\rank\ker\theta_{k}$ at a general point of $\mathbb{P}^{1}$ . Let $s$ be such a general point. Then, $\theta_{k}$ at $s$ is given by

\displaystyle(\theta_{k})_{s}

\displaystyle=\begin{pmatrix}A_{1}^{k}&B_{2}^{k+1}&&&\\ &A_{2}^{k+1}&B_{3}^{k+2}&\\ &&\ddots&\ddots\\ &&&A_{r-k}^{r-1}&B_{r-k+1}^{r}\\ \end{pmatrix}

where

	$\displaystyle A_{j}^{j+k-1}(f_{j}^{j+k})$	$\displaystyle=(\theta^{j+k})_{s}\circ f_{j}^{j+k}$
	$\displaystyle B_{j+1}^{j+k}(f_{j}^{j+k})$	$\displaystyle=-(f_{j}^{j+k}\otimes\operatorname{id})\circ(\theta^{j+1})_{s}.$

Then,

\displaystyle\dim\rank(\theta_{k})_{s}\leq\sum_{j=1}^{r-k}\dim\rank(A_{j}^{j+k-1})

and since $s$ is a general point, we know that our Higgs fields $\theta^{p}:E^{p}\to E^{p-1}\otimes\Omega_{X}^{1}(\log D)$ are nonzero and so our maps $A_{j}^{j+k-1}$ are also nonzero. Therefore

\displaystyle\rank V^{k-1}-\rank\mathrm{coker}\,\theta_{k}=\rank V^{k}-\rank\ker\theta_{k}=\dim(V^{k})_{s}-\dim\ker(\theta_{k})_{s}\geq r-k.

∎

We can now prove Proposition 5.1.

Proof of Proposition 5.1.

We have an exact sequence of sheaves

\displaystyle 0\to\ker\theta_{k}\to V^{k}\to V^{k-1}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)\to\mathrm{coker}\,\theta_{k}\to 0

and by Corollary 5.3 for $1<k<r$ this is an exact sequence of vector bundles on $\mathbb{P}^{1}$ . Therefore we obtain

\displaystyle\operatorname{par-deg}\ker\theta_{k}

\displaystyle+\operatorname{par-deg}V^{k-1}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)=\operatorname{par-deg}V^{k}+\operatorname{par-deg}\mathrm{coker}\,\theta_{k}.

Since $\operatorname{par-deg}V^{k-1}\otimes\Omega_{\mathbb{P}^{1}}^{1}(\log D)=\operatorname{par-deg}V^{k-1}+\rank V^{k-1}\cdot(\deg D-2)$ , by summing the above equation over $1<k<r$ we obtain the equality

	$\displaystyle\sum_{k=2}^{r-1}\operatorname{par-deg}\ker\theta_{k}+\sum_{k=2}^{r-1}\operatorname{par-deg}V^{k-1}+(\deg D-2)\sum_{k=2}^{r-1}\rank V^{k-1}$
	$\displaystyle=\sum_{k=2}^{r-1}\operatorname{par-deg}V^{k}+\sum_{k=2}^{r-1}\operatorname{par-deg}\mathrm{coker}\,\theta_{k}.$

Simplifying, rearranging, and applying Lemma 5.5 yields

	$\displaystyle\sum_{k=2}^{r-1}\operatorname{par-deg}\mathrm{coker}\,\theta_{k}-\sum_{k=2}^{r-1}\operatorname{par-deg}\ker\theta_{k}$
	$\displaystyle\geq(\deg D-2)\sum_{k=2}^{r-1}\rank V^{k-1}+(\deg D-1)\sum_{k=2}^{r-1}(k-2)\rank V^{k}-\sum_{k=2}^{r-1}(k-1)\rank V^{k}-\rank V^{r-1}.$

By applying Lemma 2.14 to the parabolic bundles $\ker\theta_{k}$ and $\mathrm{coker}\,\theta_{k}$ , we have

	$\displaystyle(\deg D-2)\sum_{k=2}^{r-1}\rank V^{k-1}+\operatorname{par-deg}V^{1}-\operatorname{par-deg}V^{r-1}$
	$\displaystyle<\sum_{k=2}^{r-1}\deg\mathrm{coker}\,\theta_{k}+\deg D\cdot\left(\sum_{k=2}^{r-1}\rank\mathrm{coker}\,\theta_{k}\right)-\sum_{k=2}^{r-1}\deg\ker\theta_{k}.$

Then Corollary 5.4 and the rank-nullity theorem impies

	$\displaystyle\sum_{k=2}^{r-1}\deg\mathrm{coker}\,\theta_{k}+\deg D\cdot\left(\sum_{k=2}^{r-1}\rank\mathrm{coker}\,\theta_{k}\right)-\sum_{k=2}^{r-1}\deg\ker\theta_{k}$
	$\displaystyle\leq\rank V^{r-1}-\rank V^{1}+\deg D\cdot\left(\sum_{k=2}^{r-1}\rank\mathrm{coker}\,\theta_{k}\right).$

We can now combine inequalities to get

	$\displaystyle\rank V^{r-1}-\rank V^{1}+\deg D\cdot\left(\sum_{k=2}^{r-1}\rank\mathrm{coker}\,\theta_{k}\right)$
	$\displaystyle\geq(\deg D-2)\sum_{k=2}^{r-1}\rank V^{k-1}+(\deg D-1)\sum_{k=2}^{r-1}(k-2)\rank V^{k}-\sum_{k=2}^{r-1}(k-1)\rank V^{k}-\rank V^{r-1}.$

Simplifying yields

	$\displaystyle 2\rank V^{r-1}-\rank V^{1}+2\sum_{k=2}^{r-1}\rank V^{k-1}+\sum_{k=2}^{r-1}(2k-3)\rank V^{k}$
	$\displaystyle\geq\deg D\cdot\left(\sum_{k=2}^{r-1}(\rank V^{k-1}-\rank\mathrm{coker}\,\theta_{k})+\sum_{k=2}^{r-1}(k-2)\rank V^{k}\right).$

Observe that $\rank V^{k-1}>\mathrm{coker}\,\theta_{k}$ for all $1<k<r$ , so the expression on the right-hand-side is positive by Lemma 5.6. Therefore,

\displaystyle\deg D\leq\frac{2\rank V^{r-1}-\rank V^{1}+2\sum_{k=2}^{r-1}\rank V^{k-1}+\sum_{k=2}^{r-1}(2k-3)\rank V^{k}}{\sum_{k=2}^{r-1}(\rank V^{k-1}-\rank\mathrm{coker}\,\theta_{k})+\sum_{k=2}^{r-1}(k-2)\rank V^{k}}.

∎

We can now extract a uniform bound on $\deg D$ in terms of $\rank E_{*}$ from Proposition 5.1.

Proof of Theorem 1.3.

If $(E_{*},\theta)$ has more than $2$ nonzero pieces in its grading, $(E_{*},\theta)$ is given by a Higgs bundle $(E_{*},\theta_{*})$ with nonzero Higgs field as it underlies a non-unitary local system. As our local monodromy data is generic, our parabolic weights are all distinct. And the $\theta^{p}$ are never identically zero as $\mathbb{V}$ underlies an irreducible representation. Let $r$ be the Hodge length of $(E_{*},\theta)$ .

We can apply Proposition 5.1. We have that

\displaystyle\deg D\leq\frac{2\rank V^{r-1}-\rank V^{1}+2\sum_{k=2}^{r-1}\rank V^{k-1}+\sum_{k=2}^{r-1}(2k-3)\rank V^{k}}{\sum_{k=2}^{r-1}(\rank V^{k-1}-\rank\mathrm{coker}\,\theta_{k})+\sum_{k=2}^{r-1}(k-2)\rank V^{k}}.

By Lemma 5.6, we have that $\rank V^{k-1}-\rank\mathrm{coker}\,\theta_{k}\geq r-k$ and so

\displaystyle\deg D\leq\frac{2\rank V^{r-1}-\rank V^{1}+2\sum_{k=2}^{r-1}\rank V^{k-1}+\sum_{k=2}^{r-1}(2k-3)\rank V^{k}}{\sum_{k=2}^{r-1}(r-k)+\sum_{k=2}^{r-1}(k-2)\rank V^{k}}.

The denominator is bounded below by

\displaystyle\sum_{k=2}^{r-1}(r-k)=\frac{(r-1)(r-2)}{2}

and this is nonzero whenever $r>2$ .

Then we coarsely bound the numerator from above by observing that $\rank V^{k}=\rank\operatorname{Ad}E_{*}-\sum_{j\neq k}\rank V^{j}\leq n^{2}-1-2r$ and $\rank V^{k}\geq 0$ to get

	$\displaystyle 2\rank V^{r-1}-\rank V^{1}+2\sum_{k=2}^{r-1}\rank V^{k-1}+\sum_{k=2}^{r-1}(2k-3)\rank V^{k}$
	$\displaystyle\leq 2(n^{2}-1-2r)-2\sum_{k=2}^{r-1}(n^{2}-1-2r)+\sum_{k=2}^{r-1}(2k-3)(n^{2}-1-2r)$
	$\displaystyle\leq 2(n^{2}-1-2r)-2(n^{2}-1-2r)(r-2)+2(n^{2}-1-2r)(r-2)^{2}$
	$\displaystyle=2(r^{2}-5r+7)(n^{2}-2r-1)$

For $r>2$ , we have that $r^{2}-5r+7<(r-1)(r-2)$ and so

\displaystyle\deg D\leq\frac{2(r^{2}-5r+7)(n^{2}-2r-1)}{\frac{(r-1)(r-2)}{2}}<4(n^{2}-2r-1)\leq 4n^{2}-28.

Therefore $\deg D<4n^{2}-28$ , and this contradicts our assumption that $\deg D\geq 4n^{2}-28$ . Therefore, $r\leq 2$ as desired. ∎

6. Consequences of Theorem 1.3

Theorem 1.3 states that when $\deg D\geq 4n^{2}-28$ , then a stable minimal energy parabolic Higgs bundle of rank $n$ has at most two graded pieces. We show that this is the best we can do. We construct an example of a stable minimal energy parabolic Higgs bundle on $\mathbb{P}^{1}$ in every rank with parabolic divisor of arbitrarily large degree which has two graded pieces. We first prove a technical lemma.

Lemma 6.1.

Let $a$ be a nonnegative integer, $n$ an integer larger than $1$ , and $\epsilon>0$ a very small real number. Then,

\displaystyle\frac{1}{n-1}\left(1-\frac{1+a}{1+na}-\epsilon\right)<\frac{1+a}{1+na}-\epsilon.

Proof.

We observe that

\displaystyle a+1>a+\frac{1}{n}=\frac{1}{n}\left(na+1\right)

and so

\displaystyle\left(\frac{n}{n-1}\right)\left(\frac{1+a}{1+na}\right)>\frac{1}{n-1}.

We can choose $\epsilon>0$ to be small enough so that

\displaystyle\left(\frac{n}{n-1}\right)\left(\frac{1+a}{1+na}\right)>\frac{1}{n-1}+\epsilon\left(\frac{n-2}{n-1}\right).

Then,

\displaystyle\left(\frac{n}{n-1}\right)\left(\frac{1+a}{1+na}\right)=\frac{1}{n-1}\left(\frac{1+a}{1+na}\right)+\frac{1+a}{1+na}>\frac{1}{n-1}+\epsilon-\epsilon\left(\frac{1}{n-1}\right).

Rearranging yields

\displaystyle\frac{1+a}{1+na}-\epsilon>\frac{1}{n-1}\left(1-\frac{1+a}{1+na}-\epsilon\right).

∎

Example 6.2.

Let $a$ be some nonnegative integer and $\epsilon>0$ very small. Let $E=S\oplus Q$ where $S=\mathscr{O}(-a)^{\oplus(n-1)}$ and $Q=\mathscr{O}(-a-1)$ , so $\rank E=n$ . We let $D$ be a reduced effective divisor on $\mathbb{P}^{1}$ so that $\deg D=1+na$ . We choose our Higgs field $\theta:S\to Q\otimes\Omega_{X}^{1}(\log D)$ to be generic, and since $Q$ is a line bundle we know that $\theta$ is surjective. At each $x_{j}$ , we give $E$ the weights

\displaystyle 0<\frac{1}{n-1}\left(1-\frac{1+a}{1+na}+\epsilon\right)+\epsilon_{1}<\dots<\frac{1}{n-1}\left(1-\frac{1+a}{1+na}+\epsilon\right)+\epsilon_{n-1}<\frac{1+a}{1+na}-\epsilon<1

where $\epsilon_{1}<\dots<\epsilon_{n-1}$ are chosen to be very close to zero, and so that $\epsilon_{1}+\dots+\epsilon_{n-1}=0$ . By Lemma 6.1, we can pick $\epsilon_{1},\dots,\epsilon_{n-1}$ satisfying these conditions.

We give $E_{x_{j}}$ the parabolic flag $E_{x_{j}}=S_{j}^{1}\oplus Q_{x_{j}}\supset\dots\supset S_{j}^{n-1}\oplus Q_{x_{j}}\supset Q_{x_{j}}\supseteq 0$ where $\{S_{j}^{i}\}$ is a generically chosen flag of $S_{x_{j}}$ . Our parabolic weights are distinct and our parabolic structure can be chosen to be generic in the sense of Definition 3.6 by modifying the real numbers $\epsilon$ and $\epsilon_{1},\dots,\epsilon_{n-1}$ .

We actually have a nonzero Higgs field beecause $\mathscr{H}\kern-2.0ptom(S_{*},Q_{*})_{*}\otimes\Omega_{X}^{1}(\log D)\cong\mathscr{O}(na)^{\oplus(n-1)}$ which has global sections since $n\geq 0$ and $a\geq 0$ .

Since $(E_{*},\theta)$ only has two graded pieces, the complex $\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)}$ has graded pieces

	$\displaystyle 0$	$\displaystyle\to\mathscr{H}\kern-2.0ptom(Q_{},S_{})_{*}\otimes\Omega_{X}^{1}(\log D)$
	$\displaystyle\mathscr{H}\kern-2.0ptom(Q_{},S_{})_{*}\otimes\Omega_{X}^{1}(\log D)$	$\displaystyle\to\widehat{\mathscr{H}\kern-2.0ptom(S_{},S_{})_{}\otimes\Omega_{X}^{1}(\log D)}\oplus\widehat{\mathscr{H}\kern-2.0ptom(Q_{},Q_{})_{}\otimes\Omega_{X}^{1}(\log D)}$
	$\displaystyle\mathscr{H}\kern-2.0ptom(S_{},S_{})_{}\oplus\mathscr{H}\kern-2.0ptom(Q_{},Q_{})_{}$	$\displaystyle\to\mathscr{H}\kern-2.0ptom(S_{},Q_{})_{*}\otimes\Omega_{X}^{1}(\log D)$
	$\displaystyle\mathscr{H}\kern-2.0ptom(S_{},Q_{})_{*}$	$\displaystyle\to 0.$

By symmetry of the Hodge structure on $\mathbb{H}^{1}(\operatorname{Ad}(E_{*})\to\widehat{\operatorname{Ad}(E_{*})\otimes\Omega_{X}^{1}(\log D)})$ , to show that $E_{*}$ is of minimal energy we need to show that $\mathbb{H}^{1}(\mathscr{H}\kern-2.0ptom(S_{*},Q_{*})_{*}\to 0)=H^{1}(\mathbb{P}^{1},\mathscr{H}\kern-2.0ptom(S_{*},Q_{*})_{*})=0$ . But this is true, since $\mathscr{H}\kern-2.0ptom(S_{*},Q_{*})_{*}\cong\mathscr{O}(-1)^{\oplus(n-1)}$ .

Therefore it remains to show that $(E_{*},\theta)$ is a parabolic stable Higgs bundle. By construction, the parabolic degree of $E_{*}$ is zero. The sub-Higgs bundles of $(E_{*},\theta)$ are the bundles $Q_{*}$ , $\ker(\theta)_{*}$ , and $V_{*}\oplus Q_{*}$ where $V\subseteq S$ is any subbundle.

We first note that $\mu_{*}(Q_{*})=\deg Q+\sum_{x_{j}\in D}(\frac{1+a}{1+na}-\epsilon)=-1-a+1+a-(1+na)\epsilon<0$ since $\deg D=1+na$ so $Q_{*}$ is not destabilizing.

To check that $\ker(\theta)_{*}$ is not destabilizing, we note that it fits in a exact sequence

\displaystyle 0\to\ker(\theta)_{*}\to S_{*}\to Q_{*}\otimes\Omega_{X}^{1}(\log D)\to 0.

Hence by Lemma 2.4, we have the equality

\displaystyle\operatorname{par-deg}\ker(\theta)_{*}=\operatorname{par-deg}S_{*}-\operatorname{par-deg}Q_{*}\otimes\Omega_{X}^{1}(\log D).

Observe that $\operatorname{par-deg}S_{*}=\epsilon\deg D$ and $\operatorname{par-deg}Q_{*}=-\epsilon\deg D$ , so

\displaystyle\operatorname{par-deg}\ker(\theta)_{*}=\operatorname{par-deg}S_{*}-\operatorname{par-deg}Q_{*}\otimes\Omega_{X}^{1}(\log D)=2\epsilon\deg D-\deg D+2.

Since $\epsilon>0$ is chosen to be very small, as long as $\deg D>3$ (which holds, for example, as long as $a>0$ and $n>1$ ) we know that $\operatorname{par-deg}\ker(\theta)_{*}<0$ and so $\ker(\theta)_{*}$ cannot be destabilizing.

It remains to check that $V_{*}\oplus Q_{*}$ is not destabilizing. We know that $\operatorname{par-deg}(V_{*}\oplus Q_{*})=\operatorname{par-deg}V_{*}+\operatorname{par-deg}Q_{*}=\operatorname{par-deg}V_{*}-\epsilon\deg D$ . Let $r=\rank V$ . Then,

	$\displaystyle\operatorname{par-deg}V$	$\displaystyle\leq-ra+\deg D\sum_{j=n-1-r}^{n-1}\left(\frac{1}{n-1}\left(1-\frac{1+a}{1+na}+\epsilon\right)+\epsilon_{j}\right)$
		$\displaystyle=-ra+\frac{r\deg D}{n-1}\left(1-\frac{1+a}{1+na}+\epsilon\right)+\deg D(\epsilon_{n-1}+\dots+\epsilon_{n-1-r}).$

Simplifying and substituting $\deg D=1+na$ into the bound yields

\displaystyle\operatorname{par-deg}V

\displaystyle\leq\frac{r\deg D\epsilon}{n-1}+\deg D(\epsilon_{n-1}+\dots+\epsilon_{n-1-r}).

Therefore,

\displaystyle\operatorname{par-deg}(V_{*}\oplus Q_{*})\leq\frac{r\deg D\epsilon}{n-1}+\deg D(\epsilon_{n-1}+\dots+\epsilon_{n-1-r})-\epsilon\deg D.

Since $0\leq r<n-1$ and $\epsilon_{n-1}+\dots+\epsilon_{n-1-r}$ is chosen to be very small, we have $\operatorname{par-deg}(V_{*}\oplus Q_{*})<0$ .

Therefore every sub-Higgs bundle of $(E_{*},\theta)$ fails to be destabilizing, and so $(E_{*},\theta)$ is indeed a minimal energy parabolic Higgs bundle of rank $n$ with parabolic divisor $D$ that has two graded pieces.

By choosing $a\gg 0$ , we can force $\deg D=na+1$ to be larger than $4n^{2}-28$ . Therefore, we cannot do better than two graded pieces in Theorem 1.3. In this example, we picked our parabolic weights so that at each point in $D$ , the weights sum to an integer. Therefore, the local monodromy data is not only unitary but actually lies in $\operatorname{SU}(n)$ .

6.3. An application to Gromov-Witten theory

We give an application of minimal energy local systems to Gromov-Witten invariants.

To a parabolic bundle $V_{*}$ of rank $n$ on $\mathbb{P}^{1}$ with parabolic divisor $D$ , we can obtain a modified complete parabolic bundle $\widetilde{V}_{*}$ with the following properties. This process is described in [AW98, below Theorem 5.1] and in [Bel01, Appendix]. We describe their properties below.

Proposition 6.4.

Let $V_{*}$ be a parabolic bundle and $\widetilde{V}_{*}$ its modified parabolic bundle.

(1)

The underlying bundle of $\widetilde{V_{*}}$ is a trivial bundle so that $\rank\tilde{V}=\rank V$ .
(2)

If $V_{*}$ has $0\leq\alpha_{j}^{1}<\dots<\alpha_{j}^{n_{j}}<1$ at $x_{j}\in D$ , then the weights $\{\widetilde{\alpha_{j}^{i}}\}$ of $\widetilde{V}_{*}$ at $x_{j}\in D$ no longer necessarily lie within $[0,1)$ but satisfy $\widetilde{\alpha_{j}^{1}}\leq\dots\leq\widetilde{\alpha_{j}^{n}}<\widetilde{\alpha_{j}^{1}}+1$ . The weights can be repeated.
(3)

The parabolic degree of a modified parabolic bundle is defined to be the sum of the degree of the underlying bundle and the weights above every point. Then, $\operatorname{par-deg}(V_{*})=\operatorname{par-deg}(\widetilde{V}_{*})$ .
(4)

There is a bijection between subbundles $W_{*}$ of $V_{*}$ and modified parabolic subbundles $\widetilde{W}_{*}$ of $\widetilde{V}_{*}$ . This bijection preserves parabolic degree. If $W_{*}$ and $\widetilde{W}_{*}$ are sent to each other under this bijection, then $\operatorname{par-deg}W_{*}=\operatorname{par-deg}\widetilde{W}_{*}$ .
(5)

As a consequence of (1), (3) and (4), $V_{*}$ is parabolic stable if and only if $\widetilde{V}_{*}$ is parabolic stable.

A conjugacy class $C\subseteq\operatorname{SU}(n)$ is determined by the eigenvalues of a diagonal representative. If the eigenvalues are given by $(e^{2\pi i\theta_{1}},\dots,e^{2\pi i\theta_{n}})$ , then we obtain real numbers $\theta_{1},\dots,\theta_{n}$ where each $\theta_{i}$ is determined up to an integer. Note that the $\theta_{1},\dots,\theta_{n}$ must sum to an integer because matrices in $\operatorname{SU}(n)$ have determinant one. We choose the $\theta_{1},\dots,\theta_{n}$ so that $\theta_{1}+\dots+\theta_{n}=0$ and that (after ordering from largest to smallest) $\theta_{1}\geq\dots\geq\theta_{n}\geq\theta_{1}-1$ . This uniquely determines the conjugacy class $C$ . Let $\lambda(C)=(\theta_{1},\dots,\theta_{n})$ and for some subset $I\subseteq\{1,\dots,n\}$ of size $k$ we let $\lambda_{I}(C)=\sum_{j\in I}\theta_{j}$ .

A subset $I\subseteq\{1,\dots,n\}$ of size $k$ determines a Schubert variety $\Omega_{I}$ in the Grassmannian $\operatorname{Gr}(k,n)$ , and we let $\sigma_{I}$ be the cohomology class associated to $\Omega_{I}$ . When $I_{1},\dots,I_{d}$ are all of subsets of $\{1,\dots,n\}$ of same size, we let $\langle\sigma_{I_{1}},\dots,\sigma_{I_{d}}\rangle_{\beta}$ be the number of degree $\beta$ be rational curves in $\operatorname{Gr}(k,n)$ passing through $\Omega_{I_{1}},\dots,\Omega_{I_{d}}$ . For details on Schubert varieties and Gromov-Witten invariants, see [Bel01, Definition 4].

Then, Belkale [Bel01, Theorem 7] and Agnihotri-Woodward [AW98, Theorem 3.1] independently prove the following:

Theorem 6.5.

There is a $\operatorname{SU}(n)$ -local system on $\mathbb{P}^{1}\setminus\{x_{1},\dots,x_{d}\}$ with prescribed $\operatorname{SU}(n)$ -local monodromy $C_{1},\dots,C_{d}$ if and only if for any $s\in\{1,\dots,n-1\}$ and any choice of subsets $I_{1},\dots,I_{d}\subseteq\{1,\dots,n\}$ of cardinality $s$ , the inequality

\displaystyle\sum_{k=1}^{r}\lambda_{I_{k}}(C_{k})\leq d

whenever $\langle\sigma_{I_{1}},\dots,\sigma_{I_{k}}\rangle_{d}=1$ .

The above statement is taken from [Bel01, Theorem 7].

Corollary 6.6.

Let $X=\mathbb{P}^{1}$ and $D=x_{1}+\dots+x_{d}$ be a reduced effective divisor. Let $C_{1},\dots,C_{d}\subseteq\operatorname{SU}(n)$ be a collection generic local monodromy data. Let $(E_{*},\theta)$ be a minimal energy stable parabolic Higgs bundle which only has two graded pieces and has parabolic weights corresponding to the data $C_{1},\dots,C_{d}$ . Then, $E_{*}=H_{*}\oplus(V/H)_{*}$ with Higgs field $\theta:H\to(V/H)\otimes\Omega_{X}^{1}(\log D)$ .

Let $\widetilde{E}_{*}$ be the modified parabolic bundle of $E_{*}$ and $\widetilde{H}_{*}$ the modified parabolic bundle of $H_{*}$ . For all $j=1,\dots,d$ we let $I_{j}\subseteq\{1,\dots,n\}$ be a subset of size $\rank H$ so that for all $i\in I$ , $\theta_{j}^{i}$ is a parabolic weight of $\widetilde{H}_{*}$ . Then, the Gromov-Witten invariant $\langle\sigma_{I_{1}},\dots,\sigma_{I_{d}}\rangle_{-\deg\widetilde{H}}$ is nonzero.

Proof.

If there is a minimal energy local system $\mathbb{V}$ on $\mathbb{P}^{1}\setminus D$ with the local monodromy data $C_{1},\dots,C_{d}$ which is not $\operatorname{SU}(n)$ -local systems, then there are no $\operatorname{SU}(n)$ -local systems satisfying this local monodromy data. In particular, $H_{*}$ is a subbundle of $E_{*}$ which is destabilizing as a parabolic subbundle. Let $m=\rank k$ .

We know that $\widetilde{H}_{*}$ is a destabilizing subbundle of $\widetilde{V}_{*}$ . Hence, the associated Gromov-Witten invariant $\langle\sigma_{I_{1}},\dots,\sigma_{I_{d}}\rangle_{-\deg\widetilde{H}}$ is nonzero. This is because $\widetilde{H}$ is a subbundle of a trivial bundle and hence determines a rational degree $-\deg\widetilde{H}$ curve in $\operatorname{Gr}(k,n)$ intersecting the Schubert varieties $\Omega_{I_{1}},\dots,\Omega_{I_{d}}$ . ∎

Remark 6.7.

When $\deg D\geq 4n^{2}-28$ where $n=\rank E$ , Theorem 1.3 applies and the minimal energy local systems have the desired number of graded pieces.

We give an example of Corollary 6.6.

Example 6.8.

Let $(E_{*},\theta)$ be a parabolic Higgs bundle on $\mathbb{P}^{1}$ with parabolic divisor $D=x_{1}+x_{2}+x_{3}$ . We let the underlying bundle be $E=\mathscr{O}(-2)\oplus\mathscr{O}(-1)\oplus\mathscr{O}(-1)$ . We set $S=\mathscr{O}(-2)$ and $Q=\mathscr{O}(-1)\oplus\mathscr{O}(-1)$ . We let $Q_{x_{j}}\supseteq Q_{j}\supseteq 0$ be a generic flag of the fiber $Q_{x_{j}}$ at each point $x_{j}$ . For $j=1,2$ , we let our parabolic flag at $x_{j}$ be given by

\displaystyle E_{x_{j}}=Q_{x_{j}}\oplus S_{x_{j}}\supseteq Q_{j}\oplus S_{x_{j}}\supseteq S_{x_{j}}\supseteq 0.

At $x_{3}$ , we let our parabolic flag be given by

\displaystyle E_{x_{3}}=Q_{x_{3}}\oplus S_{x_{3}}\supseteq Q_{x_{3}}\supseteq Q_{3}\supseteq 0.

Let $\frac{1}{24}>\epsilon>\frac{1}{48}$ be a real number. We choose our weights at $x_{1}$ to be $\{\frac{1}{2},\frac{1}{2}+\epsilon,1-\epsilon\}$ . The weights at $x_{2}$ are $\{\frac{1}{8}-\epsilon,\frac{1}{8},\frac{3}{4}+\epsilon\}$ . The weights at $x_{3}$ are given by $\{\frac{1}{3}-\epsilon,\frac{1}{3},\frac{1}{3}+\epsilon\}$ . That is, at $x_{1}$ and $x_{2}$ we give $S$ the largest parabolic weights. At $x_{3}$ we give $S$ the smallest parabolic weights.

We note that $\mu(S_{*})=\frac{1}{12}-\epsilon$ . Every subbundle of $Q_{*}$ has parabolic degree at most $-\frac{1}{24}-\epsilon$ . Therefore, $(E_{*},\theta)$ is parabolic stable. Then, $\operatorname{Hom}(S_{*},Q_{*})_{*}\cong\mathscr{O}(-1)\oplus\mathscr{O}(-1)$ and $\operatorname{Hom}_{*}(S_{*},Q_{*})_{*}\otimes\Omega_{X}^{1}(\log D)\cong\mathscr{O}\oplus\mathscr{O}$ . Therefore, $(E_{*},\theta)$ is a minimal energy parabolic Higgs bundle.

Let $I_{1}$ , $I_{2}$ , and $I_{3}$ be the subsets of $\{1,2,3\}$ of size $1$ so that $\lambda_{I_{j}}(C_{j})$ gives the weight of $S_{*}$ over $x_{j}$ . So, our Schubert varieties are subvarieties of $\mathbb{P}^{2}$ . Here, $I_{1}=I_{2}=\{1\}$ and $I_{3}=\{3\}$ . The classes $\sigma_{I_{1}}$ and $\sigma_{I_{2}}$ are the class of a point. The class $\sigma_{I_{3}}$ is the class of $\mathbb{P}^{2}$ and $\deg\widetilde{S}=-1$ . Therefore, we get that $\langle\sigma_{I_{1}},\sigma_{I_{2}},\sigma_{I_{3}}\rangle_{1}\neq 0$ . Geometrically, this says that there is at least one degree $1$ rational curve in $\mathbb{P}^{2}$ passing through all of $\mathbb{P}^{2}$ and two generic points in $\mathbb{P}^{2}$ . Since a degree $1$ rational curve is just a line, this says that there is at least one line passing through two points in $\mathbb{P}^{2}$ in generic position. Corollary 6.6 does not, however, tell us that there is a unique line passing through two points in $\mathbb{P}^{2}$ in generic position.

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	$\displaystyle 0$	$\displaystyle\to\mathscr{H}\kern-2.0ptom(Q_{},S_{})_{*}\otimes\Omega_{X}^{1}(\log D)$
	$\displaystyle\mathscr{H}\kern-2.0ptom(Q_{},S_{})_{*}\otimes\Omega_{X}^{1}(\log D)$	$\displaystyle\to\widehat{\mathscr{H}\kern-2.0ptom(S_{},S_{})_{}\otimes\Omega_{X}^{1}(\log D)}\oplus\widehat{\mathscr{H}\kern-2.0ptom(Q_{},Q_{})_{}\otimes\Omega_{X}^{1}(\log D)}$
	$\displaystyle\mathscr{H}\kern-2.0ptom(S_{},S_{})_{}\oplus\mathscr{H}\kern-2.0ptom(Q_{},Q_{})_{}$	$\displaystyle\to\mathscr{H}\kern-2.0ptom(S_{},Q_{})_{*}\otimes\Omega_{X}^{1}(\log D)$
	$\displaystyle\mathscr{H}\kern-2.0ptom(S_{},Q_{})_{*}$	$\displaystyle\to 0.$

Minimal Energy Local Systems on Curves

Abstract.

1. Introduction

1.1. Main results

Theorem 1.2.

Theorem 1.3.

1.4. History and motivation

1.5. Structure of the paper

1.6. Acknowledgements

2. Parabolic Bundles

Definition 2.1.

Definition 2.2.

Definition 2.3.

Lemma 2.4.

Proof.

Definition 2.5.

Definition 2.6.

Example 2.7.

Definition 2.8.

Definition 2.9.

Proposition 2.10.

Proof.

Definition 2.11.

Definition 2.12.

Lemma 2.13.

Proof.

Lemma 2.14.

Proof.

Proposition 2.15.

Proof.

3. Variations of Hodge Structure

Definition 3.1.

Definition 3.2.

Definition 3.3.

Definition 3.4.

Proposition 3.5.

Proof.

Definition 3.6.

Example 3.7.

Definition 3.8.

4. Minimal Energy

Remark 4.1.

Definition 4.2.

Remark 4.3.

Proof.

Remark 4.4.

Example 4.5.

Example 4.6.

Proposition 4.7.

Proof.

4.8. The Hitchin map

Definition 4.9.

Proposition 4.10.

Proof.

Lemma 4.11.

Proof.

4.12. The Bialynicki-Birula stratification

Theorem 4.13.

Remark 4.14.

4.15. Proof of Theorem 1.2

Proof of Theorem 1.2.

4.16. Some properties of minimal energy local systems

Proposition 4.17.

Proof.

Proposition 4.18.

Proof.

Proposition 4.19.

Proof.

5. Genus g=0g=0

Proposition 5.1.

Lemma 5.2.

Proof.

Corollary 5.3.

Proof.

Corollary 5.4.

Proof.

Lemma 5.5.

Proof.

Lemma 5.6.

Proof.

5. Genus $g=0$